Industrial Separation Processes: Fundamentals 9783110654806, 9783110654738

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André B. de Haan, H. Burak Eral, Boelo Schuur Industrial Separation Processes

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André B. de Haan, H. Burak Eral, Boelo Schuur

Industrial Separation Processes Fundamentals 2nd Edition

Authors Prof. Dr. Ir. André B. de Haan Delft University of Technology Department of Chemical Engineering Section Transport Phenomena Van der Maasweg 9 2629 HZ Delft The Netherlands Assoc. Prof. Dr. H. Burak Eral Delft University of Technology Process & Energy Department Leeghwaterstraat 39 2628 CB Delft The Netherlands Prof. Dr. Ir. Boelo Schuur Sustainable Process Technology group Faculty of Science and Technology University of Twente Meander building 221 De Horst 2 7522 LW Enschede The Netherlands

ISBN 978-3-11-065473-8 e-ISBN (PDF) 978-3-11-065480-6 e-ISBN (EPUB) 978-3-11-065491-2 Library of Congress Control Number: 2020932665 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2020 Walter de Gruyter GmbH, Berlin/Boston Cover image: RonFullHD/iStock/Getty Images Typesetting: Integra Software Services Pvt. Ltd. Printing and binding: CPI books GmbH, Leck www.degruyter.com

Preface This textbook has originally been written to support the bachelor course “Separation Technology” at the University of Twente and has been later also used at Eindhoven University of Technology, Delft University of Technology, Hogeschool Rotterdam, and other institutions of higher education. Our main objective is to present an overview of the fundamentals underlying the most frequently used industrial separation methods. We focus on their physics principles and the basic computation methods that are required to assess their technical and economic feasibility for a given application. Thus, design calculations are limited to those required for the development of conceptual process schemes. To keep computational time within the limits of our course structure, most examples and exercises of homogeneous mixtures are limited to binary mixtures. The textbook is organized into three main parts. Separation processes for homogeneous mixtures are treated in the parts on equilibrium-based molecular separations (Chapters 2–5) and rate-controlled molecular separations (Chapters 6–8). Mechanical separation technology presented in Chapters 9–11 provides an overview of the most important techniques for the separation of heterogeneous mixtures. In each chapter a short overview of the most commonly used equipment types is given. Only for gas–liquid contactors Chapter 4 goes into more detail about their design and operation because they are the most commonly used industrial contactors. Chapter 12 has a unique position as this chapter considers the selection of an appropriate separation process for a given separation task. The design of separation processes can only be learned by an active hands-on approach. The most important aspects are the combination of the right material balances with the thermodynamic equilibrium relations and mass transport equations. To support the reader in learning and applying the presented material, we have extended the number of exercises included at the end of each chapter. Short answers are given at the end of this book; detailed solutions are given in a separate solution manual that is available at the website (http://dx.doi.org/10.1515/9783110306729_suppl.). In the preparation of 2nd edition of this book, we greatly appreciate Hans Bosch, coauthor of the 1st edition, for allowing us to use his material from the 1st edition. Furthermore Johan Smit (Hogeschool Rotterdam) and Frederico Marques Penha (TU Delft) provided valuable comments and suggestions. André B. de Haan H. Burak Eral Boelo Schuur

https://doi.org/10.1515/9783110654806-202

Contents Preface

V

Chapter 1 Characteristics of Separation Processes 1 1.1 Significance of separations 1 1.2 Characteristics of separation processes 1.2.1 Categorization 4 1.2.2 Separating agents 5 1.2.3 Separation factors 7 1.3 Industrial separation methods 8 1.3.1 Exploitable properties 8 1.3.2 Important molecular separations 10 1.4 Inherent selectivities 11 1.4.1 Equilibrium-based processes 11 1.4.2 Rate-controlled processes 13 Nomenclature 14 Exercises 14

4

Chapter 2 Evaporation and Distillation 17 2.1 Separation by evaporation 17 2.1.1 Introduction 17 2.1.2 Vapor–liquid equilibria 18 2.2 Separation by single-stage partial evaporation 2.2.1 Differential distillation 23 2.2.2 Flash distillation 24 2.2.2.1 Specifying T and L/V 26 26 2.2.2.2 Specifying T and Ptot 2.3 Multistage distillation 28 2.3.1 Distillation cascades 28 2.3.2 Column distillation 29 2.3.3 Feasible distillation conditions 31 2.3.4 External column balances 31 2.4 McCabe–Thiele analysis 32 2.4.1 Internal balances 32 2.4.1.1 Rectifying section 33 2.4.1.2 Stripping section 34 2.4.1.3 Feed stage considerations 35 2.4.1.4 Feed line 36 2.4.2 Required number of equilibrium stages 37

23

VIII

2.4.2.1 2.4.2.2 2.4.2.3 2.4.2.4 2.4.3 2.5 2.5.1 2.5.2 2.5.3

Contents

Graphical determination of stages and location of feed stage Limiting conditions 40 Fenske and Underwood equations 41 Use of Murphree efficiency 44 Energy requirements 45 Advanced distillation techniques 45 Batch distillation 45 Continuous separation of multiple product mixtures 47 Separation of azeotropes 48 Nomenclature 50 Exercises 51

Chapter 3 Absorption and Stripping 57 3.1 Introduction 57 3.2 The aim of absorption 59 3.3 General design approach 60 3.4 Absorption and stripping equilibria 61 3.4.1 Gaseous solute solubilities 61 3.4.2 Minimum absorbent flow 63 3.5 Absorber and stripper design 64 3.5.1 Operating lines for absorption 64 3.5.2 Stripping analysis 68 3.5.3 Analytical Kremser solution 69 3.6 Industrial absorbers 72 3.6.1 Packed columns 73 3.6.2 Plate columns 74 3.6.3 Spray and bubble columns 74 3.6.4 Comparison of absorption columns 75 Nomenclature 77 Exercises 77 Chapter 4 General Design of Gas/Liquid Contactors 81 4.1 Introduction 81 4.2 Modeling mass transfer 81 4.3 Plate columns 86 4.3.1 Dimensioning a tray column 89 4.3.2 Height of a tray column 90 4.3.2.1 Tray spacing 90 4.3.2.2 Tray efficiency 90 4.3.2.3 Transfer units 92

37

IX

Contents

4.3.2.4 4.3.3 4.3.3.1 4.3.3.2 4.3.3.3 4.3.3.4 4.3.3.5 4.4 4.4.1 4.4.2 4.4.3 4.4.4 4.4.4.1 4.4.5 4.4.6 4.5

Overall efficiency 93 Diameter of a tray column 95 Flooding 96 Downcomer flooding 97 Entrainment flooding 99 Tray diameter and pressure drop 100 Remark on flooding charts 100 Packed columns 101 Random packing 102 Structured packing 103 Dimensioning a packed column 104 Height of a packed column 105 Transfer units 105 Minimum column diameter 108 Pressure drop 110 Criteria for column selection 111 Nomenclature 112 Exercises 113

Chapter 5 Liquid–Liquid Extraction 117 5.1 Introduction 117 5.1.1 General Introduction 117 5.1.2 Liquid–liquid equilibria 120 5.1.3 Solvent selection 121 5.2 Extraction schemes 123 5.2.1 Single equilibrium stage 124 5.2.2 Cocurrent cascade 125 5.2.3 Crosscurrent 126 5.2.4 Countercurrent 126 5.3 Design of countercurrent extractions 127 5.3.1 Graphical McCabe–Thiele method for immiscible systems 5.3.2 Analytical Kremser method for immiscible systems 130 5.3.3 Graphical method for partially miscible systems 131 5.3.4 Efficiency of an ideal nonequilibrium mixer 136 5.4 Industrial liquid–liquid extractors 137 5.4.1 Mixer–settlers 137 5.4.2 Mechanically agitated columns 138 5.4.3 Unagitated and pulsed columns 139 5.4.4 Centrifugal extractors 141

127

X

5.4.5

Contents

Selection of an extractor Nomenclature 147 Exercises 147

143

Chapter 6 Adsorption and Ion Exchange 155 6.1 Introduction 155 6.2 Adsorption fundamentals 158 6.2.1 Industrial adsorbents 158 6.2.2 Equilibria 164 6.2.3 Kinetics 168 6.3 Fixed-bed adsorption 171 6.3.1 Bed profiles and breakthrough curves 171 6.3.2 Equilibrium theory model 173 6.3.3 Modeling of mass transfer effects 175 6.4 Basic adsorption/desorption cycles 177 6.4.1 Temperature swing 179 6.4.2 Pressure swing 180 6.4.3 Inert and displacement purge cycles 181 6.5 Principles of ion exchange 182 6.5.1 Ion exchange resins 183 6.5.2 Equilibria and selectivity 184 6.6 Ion exchange processes 185 Nomenclature 187 Exercises 188 Chapter 7 Drying of Solids 195 7.1 Introduction 195 7.2 Humidity of carrier gas 196 7.2.1 Definitions 196 7.2.2 Air–water system: a special case 7.2.3 Psychrometric charts 199 7.3 Moisture in solids 201 7.3.1 Bound and unbound water 201 7.4 Drying mechanisms 206 7.4.1 Constant drying rate 207 7.4.2 Falling drying rate 208 7.4.3 Estimation of drying time 208 7.5 Classification of drying operations 7.5.1 Direct-heat dryers 211 7.5.1.1 Batch compartment dryers 212

198

210

Contents

7.5.1.2 7.5.1.3 7.5.1.4 7.5.1.5 7.5.1.6 7.5.2 7.5.2.1 7.5.2.2 7.5.2.3 7.5.3

Belt dryers 212 Rotary dryers 213 Flash dryers 214 Fluidized bed dryers 214 Spray dryers 216 Contact dryers 217 Rotary and agitator dryers 217 Vacuum dryers 217 Fluid bed dryers 218 Other drying methods 219 Nomenclature 220 Exercises 221 Appendix 223

Chapter 8 Crystallization 227 8.1 Introduction 227 8.2 Crystal quality parameters 232 8.2.1 Crystal size distribution 233 8.2.2 Morphology 234 8.2.3 Crystal form 235 8.2.4 Purity 236 8.2.5 Interdependency among crystal quality parameters 237 8.3 Phase diagrams 237 8.3.1 Predicting solubility curves 241 8.3.2 Measuring solubility curves 241 8.4 Kinetics of crystallization 242 8.4.1 Metastable zone limit 242 8.4.2 Measuring metastable zone width 244 8.4.3 Supersaturation 244 8.4.4 Theory of crystallization 245 8.4.5 Nucleation 247 8.4.6 Crystal growth 252 8.4.7 Population balance equations for the MSMPR crystallizer 255 8.5 Methods of supersaturation generation 259 8.5.1 Cooling 259 8.5.2 Evaporation 260 8.5.3 Vacuum crystallization 260 8.5.4 Precipitation 261 8.5.5 Anti-solvent crystallization 262 8.5.6 Melt crystallization 263

XI

XII

8.5.7 8.6 8.6.1 8.6.2 8.6.3 8.6.3.1 8.6.3.2 8.6.3.3 8.6.3.4 8.6.3.5 8.6.3.6 8.6.3.7 8.6.3.8 8.6.3.9 8.6.3.10 8.7 8.7.1 8.7.2 8.7.3 8.7.4 8.8 8.8.1 8.8.2

Contents

Choosing a method of supersaturation generation 263 Basic design for industrial crystallizers 264 Process and design specifications 266 Thermodynamic and lab-scale kinetic information 266 Design decisions 267 Selecting method of supersaturation generation 267 Operation modes: Batch versus Continuous operation 268 Number of stages: single- and multistage operation selection Selecting the crystallizer type 270 Stirred tank and cooling disk crystallizer 271 Forced circulation crystallizers 272 Draft tube baffle crystallizer 273 Fluidized bed crystallizers 273 Melt crystallizers 274 Vacuum crystallizers 276 Crystallizer calculations 277 Mass balance 277 Basic yield calculations 278 Enthalpy balance 279 Crystallizer dimensions 280 Final design procedure 281 Economic evaluation 281 Detailed design with production scale kinetic data 282 Nomenclature 283 Exercises 284

Chapter 9 Sedimentation and Settling 289 9.1 Introduction 289 9.2 Gravity sedimentation 290 9.2.1 Sedimentation mechanisms 291 9.2.2 Dilute sedimentation 293 9.2.3 Hindered settling 296 9.2.4 Continuous sedimentation tank (gravity-settling tank) 9.2.5 Gravity sedimentation equipment 299 9.2.6 Brownian motion and its influence on sedimentation 9.2.7 Colloidal stability 302 9.3 Centrifugal sedimentation 303 9.3.1 Particle velocity in a centrifugal field 304 9.3.2 Sedimenting centrifuges 304 9.3.3 Bowl centrifuge separation capability 307

296 300

270

Contents

9.3.4 9.3.5 9.3.6 9.4 9.4.1 9.4.2

Chapter 10 Filtration 10.1 10.2 10.2.1 10.2.2 10.2.3 10.2.4 10.3 10.3.1 10.3.2 10.3.3 10.4 10.5 10.5.1 10.5.2 10.6 10.6.1 10.6.2 10.6.3 10.6.4

The sigma concept 308 Capacity of disk centrifuges 310 Hydrocyclones 311 Electrostatic precipitation 315 Principles 315 Equipment and collecting efficiency Nomenclature 319 Exercises 320

317

325 The filtration process 325 Filtration fundamentals 328 Flow through packed beds 328 Cake filtration 329 Constant pressure and constant rate filtration Compressible cakes 332 Filtration equipment 333 Continuous large-scale vacuum filters 334 Batch vacuum filters 337 Pressure filters 338 Filter media 340 Centrifugal filtration 342 Centrifugal filters 342 Filtration rates in Centrifuges 344 Interceptive filtration 347 Deep bed filtration 347 Impingement filtration of gases 348 Interception mechanisms 350 Lamellar plate separators 352 Nomenclature 353 Excercises 353

Chapter 11 Membrane Filtration 359 11.1 Introduction 359 11.2 Membrane selection 360 11.3 Membrane filtration processes 362 11.4 Flux equations and selectivity 365 11.4.1 Flux definitions 366 11.4.2 Permeability for diffusion in porous membranes

331

367

XIII

XIV

11.4.3 11.4.4 11.5 11.6 11.6.1 11.7

Contents

Permeability for solution-diffusion in dense membranes Selectivity and retention 370 Concentration polarization 372 Membrane modules 375 Design procedure 379 Concluding remarks 380 Nomenclature 381 Exercises 382 Appendix 384

Chapter 12 Separation Method Selection 385 12.1 Introduction 385 12.1.1 Industrial separation processes 385 12.1.2 Factors influencing the choice of a separation process 12.1.2.1 Economics 387 12.1.2.2 Feasibility 388 12.1.2.3 Product stability 389 12.1.2.4 Design reliability 389 12.2 Selection of feasible separation processes 390 12.2.1 Classes of processes 390 12.2.2 Initial screening 391 12.2.3 Separation factor 392 12.3 Separation of homogeneous liquid mixtures 394 12.3.1 Strengths of distillation 394 12.3.2 Limitations of distillation 397 12.3.3 Low relative volatilities 399 12.3.4 Overlapping boiling points 401 12.3.5 Low concentrations 402 12.3.6 Dissolved solids 403 12.4 Separaton systems for gas mixtures 404 12.4.1 General selection considerations 404 12.4.2 Comparison of gas separation techniques 406 12.4.2.1 Absorption 406 12.4.2.2 Adsorption 407 12.4.2.3 Membrane separation 407 12.5 Separation methods for solid-liquid mixtures 408 Excercises 410

369

387

Contents

Appendix Answers to Exercises References and Further Reading Index

429

413 425

XV

Chapter 1 Characteristics of Separation Processes 1.1 Significance of separations When ethanol is placed in water, it dissolves and tends to form a solution of uniform composition. There is no simple way to separate the ethanol and the water again. This tendency of substances to form a mixture is a spontaneous, natural process that is accompanied by an increase in entropy or randomness. In order to separate the obtained mixture into its original species, a device, system or process must be used that supplies the equivalent of thermodynamic work to induce the desired separation. The fact that naturally occurring processes are inherently mixing processes makes the reverse procedure of “unmixing” or separation processes one of the most challenging categories of engineering problems. We can now define separation processes as Those Operations that Transform a Mixture of Substances into Two or More Products that Differ from Each Other in Composition

The separation of mixtures, including enrichment, concentration, purification, refining and isolation are of extreme importance to chemists and chemical engineers. Separation technology has been practiced for millennia in the food, material processing, chemical and petrochemical industry. In the food, material processing and petrochemical industry, most processes do not involve a reaction step but are merely used for the recovery and purification of products from natural resources. Examples are oil refining (Figure 1.1), metal recovery from ores and the isolation and purification of sugar from sugar beets or sugar cane schematically shown in Figure 1.2. In this process a combination of separation techniques (washing, extraction, pressing, drying, clarification, evaporation, crystallization and filtration) are used to isolate the desired product, sugar, in its right form and purity. Within the chemical industry, separation technology is mostly used in conjunction with chemical reactors. One of the main characteristics of all these chemical processes is that many more separation steps are used compared to the amount of chemical reaction steps. An illustrative example is the oxidation of ethylene to ethylene oxide shown in Figure 1.3. After the reaction, several absorption, desorption and distillation steps are required to obtain the ethylene oxide as a pure product. The above examples serve to illustrate the importance of separation operations in the majority of industrial chemical processes. Looking at the flow sheets in Figures 1.2 and 1.3 it should not be surprising that separation processes account for 40–90% of capital and operating costs in industry and their proper application can significantly reduce costs and increase profits. As illustrated by Figure 1.4, separation operations are employed to serve a variety of functions: https://doi.org/10.1515/9783110654806-001

2

Chapter 1 Characteristics of Separation Processes

Clean fuel gas LPG Sulfur Gasoline Diesel Crude oil

Oil refinery

Kerosine Lubricants Waxes Fuel oils Coke Asphalt

Figure 1.1: Very simple schematic of a refinery for converting crude oil into products.

Figure 1.2: Processing sequence for producing sugar from sugar beets.

Figure 1.3: Ethylene oxide production by oxygen oxidation.

1.1 Significance of separations

3

4

Chapter 1 Characteristics of Separation Processes

Figure 1.4: Schematic of the functions of separations in manufacturing.

– – – – – –

Removal of impurities from raw materials and feed mixtures Recycling of solvents and unconverted reactants Isolation of products for subsequent purification or processing Purification of products, product classes and recycle streams Recovery and purification of by-products Removal of contaminants from air and water effluents

1.2 Characteristics of separation processes 1.2.1 Categorization There are many examples where a separation of a heterogeneous feed consisting of more than one phase is desired. In such cases it is often beneficial to first use some mechanical means based on gravity, centrifugal force, pressure reduction or an electric and/or magnetic field to separate the phases. Such processes are called mechanical separation processes. For example, a filter of a centrifuge serves to separate solid and liquid phases from slurry. Vapor–liquid separators segregate vapor from liquid. Then, appropriate separation techniques can be applied to each phase. Most other separation processes deal with single, homogeneous mixtures (solid, liquid or gas) and involve a diffusion transfer of material from the feed stream to one of the product streams. Often a mechanical separation is employed to separate the product phases in one of these processes. These operations are referred to as

1.2 Characteristics of separation processes

5

molecular separation processes. Most molecular separation processes operate through equilibration of two immiscible phases, which have different compositions at equilibrium. Examples are evaporation, absorption, distillation and extraction processes. We shall call these equilibrium-based processes. In another class of molecular separation processes the efficiency is mainly controlled by the transport rate through media resulting from a gradient in partial pressure, temperature, composition, electric potential or the like. They are called rate-controlled processes such as adsorption, ion exchange, crystallization and drying. Important in their design is the factor time that determines in combination with the transport rates their optimal, usually cyclic way of operation.

1.2.2 Separating agents A simplified scheme of a separation process is shown in Figure 1.5. The feed mixture may consist of one or several streams and can be vapor, liquid or solid. From the fundamental nature of separation, there must be at least two product streams, which differ from each other in composition. If the feed mixture is a homogeneous solution a second phase must be developed before separation of chemical species can be achieved. This second phase can be created by an energy-separating agent (ESA), mass-separating agent (MSA) and barrier or external fields as shown in Figure 1.6. The most common industrial technique, Figure 1.6a, is the application of an energy-separating agent to create a second phase (vapor, liquid or solid) that is immiscible with the feed phase. Such applications of an energy-separating agent involve energy (heat) transfer to or from the mixture. For example, in evaporation the separating agent is the heat (energy) supplied that causes the formation of a second (vapor) phase. A vapor phase may also be created by pressure reduction. A second technique is to create a second phase in the system by the introduction of a

Figure 1.5: General separation process.

6

Chapter 1 Characteristics of Separation Processes

Phase 1

Phase 1 Feed

Energy separating agent

Feed

Phase creation

(a)

Mass separating agent

Phase 2

Phase 2

(b)

Phase 1 Phase 2

Feed

(c) Phase 1

Phase 1

Feed Barrier

(d)

Feed

Phase 2

Force field or gradient (e)

Phase 2

Figure 1.6: General separation techniques (a) by phase creation, (b) solvent addition, (c) solid mass-separating agent, (d) by barrier and (e) by force of field gradient.

mass-separating agent. Mass-separating agents may be used in the form of a liquid (absorption, extractive distillation and extraction) or a solid (adsorption and ion exchange). In extraction processes, Figure 1.6b, the separating agent is a solvent that selectively dissolves some of the species in the feed mixture. Of growing importance are techniques that involve the addition of solid particles, Figure 1.6c, which selectively adsorb solutes on their internal surface to create separation. Some separation processes utilize more than one separating agent. An example is extractive distillation, where a mixture of components with close boiling points is separated by adding a solvent (mass-separating agent), which serves to enhance the volatility of some components stronger than others, and heat (energy separating agent) in a distillation scheme to generate a more volatile product and a less volatile product. Less common is the use of a barrier, Figure 1.6d, which restricts the movement of certain chemical species with respect to other species. External fields, Figure 1.6e, of various types are sometimes applied for specialized separations. The most common industrial technique, Figure 1.6a, is the application of an energy-separating agent to create a second phase (vapor, liquid or solid) that is immiscible with the feed phase. Such applications of an energy-separating agent

1.2 Characteristics of separation processes

7

involve energy (heat) transfer to or from the mixture. For example, in evaporation the separating agent is the heat (energy) supplied that causes the formation of a second (vapor) phase. A vapor phase may also be created by pressure reduction. A second technique is to create a second phase in the system by the introduction of a mass-separating agent. Mass-separating agents may be used in the form of a liquid (absorption, extractive distillation and extraction) or a solid (adsorption and ion exchange). In extraction processes, Figure 1.6b, the separating agent is a solvent that selectively dissolves some of the species in the feed mixture. Of growing importance are techniques that involve the addition of solid particles, Figure 1.6c, which selectively adsorb solutes on their internal surface to create separation. Some separation processes utilize more than one separating agent. An example is extractive distillation, where a mixture of components with close boiling points is separated by adding a solvent (mass-separating agent), which serves to enhance the volatility of some components stronger than others, and heat (energy separating agent) in a distillation scheme to generate a more volatile product and a less volatile product. Less common is the use of a barrier, Figure 1.6d, which restricts the movement of certain chemical species with respect to other species. External fields, Figure 1.6e, of various types are sometimes applied for specialized separations. For all of the general techniques of Figure 1.6, the separations are achieved by enhancing the rate of mass transfer of certain species relative to all species. The driving force and direction of mass transfer by diffusion is governed by thermodynamics, with the usual limitations of equilibrium. Thus both transport and thermodynamic considerations are crucial in the design of separation operations. Mass transfer governs the rate of separation, while the extent of separation is limited by thermodynamic equilibrium.

1.2.3 Separation factors An important consideration in the selection of feasible separation methods is the degree of separation that can be obtained between two key components of the feed. Since the object of a separation device is to produce products of differing compositions, it is logical to define this degree of separation in terms of product compositions: This is commonly done through the separation factor for the separation of component A from component B between phases 1 and 2, defined for a single stage of contacting as SFA;B ¼

CA;1 =CA;2 xA;1 =xA;2 ¼ CB;1 =CB;2 xB;1 =xB;2

(1:1)

where C and x are composition variables, such as mole fraction, mass fraction or concentration. The value of the separation factor is limited by thermodynamic equilibrium, except in the case of membrane separations that are controlled by

8

Chapter 1 Characteristics of Separation Processes

relative rates of mass transfer through the membrane. An effective separation is accomplished when the separation factor is significantly different from unity. If SF > 1 component A tends to concentrate in the product 1 while component B accumulates in product 2. On the other hand, if SF < 1, component B tends to concentrate preferentially in product 1. In general components A and B are designated in such a manner that the separation factor is larger than unity. Consequently, the larger the value of the separation factor is, the more feasible the particular operation. In real processes the separation factor can reflect the differences in equilibrium compositions and transport rates as well as the construction and flow configuration of the separation device. For this reason it is convenient to define in Section 1.4 an inherent separation factor, the selectivity, which would be obtained under idealized conditions. For equilibrium separation processes the selectivity corresponds to those product compositions, which will be obtained when simple equilibrium is attained between the product phases. For rate-controlled separation processes the selectivity represents those product compositions, which will occur in the presence of the underlying physical transport mechanism alone. Complications from competing transport phenomena, flow configurations or other extraneous effects are excluded.

1.3 Industrial separation methods 1.3.1 Exploitable properties The objective of the design of a separation process is to exploit property differences in the most economical manner to obtain the required separation factors. Mechanical separations exploit the property differences between the coexisting phases by applying a certain force to the heterogeneous mixture. Table 1.1 presents an overview of the different mechanical separations classified according to the principle involved. The achieved extend of separation in molecular separations depends on the exploitation of differences in molecular, thermodynamic and transport properties of the different chemical species present in the feed. Some important bulk thermodynamic and transport properties are vapor pressure, adsorptivity, solubility and diffusivity. These differences in bulk properties result from differences in properties of the molecules themselves such as Molecular weight Van der Waals volume Van der Waals area Molecular shape Dipole moment

Polarizability Dielectric constant Electric charge Radius of gyration

Table 1.2 indicates the importance of the main molecular properties in determining the value of the separation factor for various separation processes. Classification of

1.3 Industrial separation methods

9

Table 1.1: Separation principles of mechanical separations. Technique

Applied mechanical force

Sedimentation Gravity Ch  Centrifugal

Electrostatic Magnetic Filtration Gravity Ch  & Ch  Pressure

Centrifugal

Impingement

Technique

Settlers Classifiers Centrifuges Cyclones Decanters Electrostatic precipitators Liquid + Solid Sieves Filtration Filtration Presses Sieves Membranes Centrifuges

Filters Scrubbers Impact separators

Applicable for particles (micron)

Separation principle

>

Density difference

–,

Density difference

.–

Charge on fine solid particles Attraction by magnetism

>

.–,

–,

.–,

Particle size larger than pore size of filter medium Particle size larger than pore size of filter medium Particle size larger than pore size of filter medium Size difference

separation processes in terms of the molecular properties that primarily govern the separation factor can be quite useful for the selection of candidate processes for separating a given mixture. Processes that emphasize molecular properties in which the components differ to the greatest should be given special attention. Although this categorization only provides general guidelines, the basic differences in the importance of different molecular properties in determining the separation factors for different separation processes are apparent from Table 1.2. For example, the separation factor in distillation reflects vapor pressures, which in turn reflect primarily the strength of intermolecular forces. The separation factor in crystallization, on the other hand, reflects primarily the ability of molecules of different kinds to fit together, and simple geometric factors of size and shape become much more important. Values of these properties for many substances are available in handbooks, specialized reference books and journals. Many of them can also be estimated using computer-aided process simulation programs. When not available, these properties must be estimated or determined experimentally if a successful application of the appropriate separation operation(s) is to be achieved.

10

Chapter 1 Characteristics of Separation Processes

Table 1.2: Relation of the separation factor to difference in molecular properties.

Distillation Crystallization Extraction Absorption Adsorption Membrane filtration Ion exchange Drying

Molecular weight

Molecular Molecular size shape

Dipole moment and polarizability

Molecular charge

Chemical reaction

++ + + + + –

+ ++ + + ++ ++

– ++ + ++ ++ +++

++ + ++ +++ ++ –

– ++ – – – –

– – ++

– ++

– +-

– –

– +

– –

+++ –

++ –

1.3.2 Important molecular separations An overview of the most commonly used industrial molecular separations divided into equilibrium and rate-based separations is given in Table 1.3. The table shows the phases involved, the separating agent and the physical or chemical principle on which the separation is based. It should be noted that the feed to a separation unit usually consists of a single vapor, liquid or solid phase. If the feed comprises two or more coexisting phases, it should be considered to separate the feed stream first into two phases by some mechanical means and then sending the separated phases to different separation units. Table 1.3 is not intended to be complete but serves as an overview of the most widely applied industrial separations. Some separation operations are well understood and can be readily designed for a mathematical model and/or scaled up to a commercial size from laboratory data. This technological and use maturity has been analyzed and is graphically illustrated in Figure 1.7 for the molecular separation technologies from Table 1.3. At the “technological asymptote” it is assumed that everything is known about the process and no further improvements are possible. If a process is used to the fullest extent possible, it has approached its “use asymptote.” As expected, the degree to which a separation operation is technologically mature correlates well with its commercial use. Those operations ranked near the top are frequently designed without the need for any laboratory of pilot-plant test. Operations near the middle usually require laboratory data, while operations near the bottom require extensive research before they can be designed. Distillation, the most frequently used process, is closer to its technological and use asymptotes than any other process.

1.4 Inherent selectivities

11

Table 1.3: Commonly used molecular separation methods. Separation method

Chapter Phase of the feed

Separation agent

Products

Separation principle

Equilibrium-based processes

Absorption

 Liquid and/or vapor  Liquid and/or vapor  Vapor

Stripping

 Liquid

Flash Distillation

Extractive distillation

Liquid and/or vapor

Azetropic distillation

Liquid and/or vapor

Extraction

 Liquid

Heat transfer

Vapor + liquid Difference in volatility Heat transfer Vapor + liquid Difference in volatility Liquid absorbent Liquid + vapor Difference in volatility Vapor stripping Vapor + liquid Difference in agent volatility Liquid solvent Vapor + liquid Difference in and heat volatility transfer Liquid entrainer Vapor + liquid Difference in and heat volatility transfer Liquid solvent Liquid + liquid Difference in solubility

Rate-controlled processes Gas adsorption Liquid adsorption Ion exchange Leaching

 Vapor

Solid adsorbent

Gas + solid

 Liquid

Solid adsorbent

Liquid + solid

 Liquid

Ion exchanger

Liquid + solid

Liquid solvent

Liquid + solid

Solid

Drying

 Solid

Heat transfer

Vapor + solid

Crystallization

 Liquid

Heat transfer

Liquid + solid

Difference in adsorbability Difference in adsorbability Difference in chemical reaction Difference in solubility Difference in volatility Difference in solubility or melting point

1.4 Inherent selectivities 1.4.1 Equilibrium-based processes For separation processes based upon the equilibration of immiscible phases the selectivity is always based on the equilibrium ratio of the components, defined as

12

Chapter 1 Characteristics of Separation Processes

Figure 1.7: Technological and use maturities of separation processes.

the ratio of the mole fraction (concentration) in phase 1 to the mole fraction in phase 2, at equilibrium: xA;1 (1:2) KA ≡ xA;2 Introduction in the previous equation for the separation factor SF, defined in eq. (1.1), provides the relation that defines the inherent selectivity SAB in terms of the mole fractions as well as equilibrium ratios: SAB =

KA xA;1 =xA;2 = KB xB;1 =xB;2

(1:3)

For processes based on equilibration between gas and liquid phases the selectivity can be related to vapor pressures P0 and activity coefficients γ: pA = yA Ptot = γA xA PA0

or KA ≡

yA γA PA0 = xA Ptot

(1:4)

For a mixture that forms nearly ideal liquid solutions Raoult’s law applies ( γ = 1). Then, the inherent selectivity for vapor–liquid separation operations employing energy as the separating agent such as partial evaporation, partial condensation or distillation is given by the following: SAB = αAB =

PA0 PB0

(1:5)

1.4 Inherent selectivities

13

The inherent selectivity αAB in a vapor–liquid system is commonly called the relative volatility. In case of nonideal solutions in vapor–liquid separation processes, the expressions for the K values of the key components must be corrected by the liquidphase activity coefficients according to eq. (1.4) and the relative volatility becomes αAB =

γA PA0 γB PB0

(1:6)

Noting that thermodynamic equilibrium requires equal activities of the components in both phases, the distribution coefficient KA for equilibrium between two nonideal immiscible liquid phases 1 and 2 follows from the introduction of the activity aA = γA xA in eq. (1.2): KA ≡

xA, 1 aA;1 =γA;1 γA, 2 = = xA, 2 aA;2 =γA;2 γA, 1

(1:7)

Hence now the inherent selectivity, for liquid–liquid extraction commonly denoted as βAB, becomes SAB = βAB =

γA;2 =γA;1 γB;2 =γB;1

(1:8)

1.4.2 Rate-controlled processes In adsorption processes a dynamic equilibrium is established for the distribution of the solute between the fluid and the solid surface. In Chapter 6, It is shown that at low amounts adsorbed, the isotherm should approach a linear form and the following form of Henry’s law: qA = HA pA

or

qA = HA0 cA

(1:9)

where q is the equilibrium loading or amount adsorbed per unit mass of adsorbent, p is the partial pressure of a gas and c is the concentration of the solute. The inherent selectivity of the adsorbent is determined similarly to the inherent selectivity defined in the previous section. For small amounts of adsorbed material, the inherent adsorption selectivity simply corresponds to the ratio of the equilibrium Henry constants HA/HB (or H’A/H’B) which equals the ratio of adsorption constants bA/bB (see eq. (6.4)): SAB =

qA =pA HA qmax · bA bA = = = qB =pB HB qmax · bB bB

(1:10)

14

Chapter 1 Characteristics of Separation Processes

The inherent selectivity for permeation through a barrier is defined in a similar way. By definition, the flux of a component through a membrane is proportional to its permeability, PM (see Chapter 11). Under ideal circumstances the inherent selectivity of permeation through a barrier is then estimated by the ratios of the permeabilities PM,A and PM,B: SAB =

PM, A PM, B

(1:11)

Nomenclature a b cA CA,i KA p Po Ptot PM Q SAB SF xA,i x, y αAB βAB γ

Activity Adsorption constant Concentration of component A Concentration of component A in phase i Equilibrium ratio (eq. (.)) Partial pressure Vapor (saturation) pressure Total pressure Permeability Equilibrium amount adsorbed Inherent selectivity (eq. .) Separation factor (eq. (.)) Mole fraction of component A in liquid phase I Mole fraction of A in liquid and vapor phases, respectively Relative volatility (eq. (.)) (=SAB for ideal vapor–liquid systems) Inherent selectivity in liquid–liquid extraction (eq. (.)) Activity coefficient

– see Ch.  mol m− mol m− – – N m− N m− see Ch  see Ch  – – – – – – –

Exercises 1 Compare and discuss the advantages and disadvantages of making separations using an energy separating agent (ESA) versus using a mass-separating agent (MSA). 2 The system benzene–toluene adheres closely to Raoult’s law. The vapor pressures of benzene and toluene at 121 °C are 300 and 133 kPa. Calculate the relative volatility. 3 As a part of the life support system for spacecraft it is necessary to provide a means of continuously removing carbon dioxide from air. It is not possible to rely upon gravity in any way to devise a CO2–air separation process.

Exercises

15

Suggest at least two separation schemes, which could be suitable for continuous CO2 removal from air under zero-gravity conditions. 4 Gold is present in seawater to a concentration level between 10−12 and 10−8 weight fraction, depending upon the location. Briefly evaluate the potential for recovering gold economically from seawater. 5 Assuming that the membrane characteristics are not changed, will the product– water purity in a reverse-osmosis seawater desalination process increase, decrease or remain constant as the upstream pressure increases. 6 Propylene and propane are among the light hydrocarbons produced by thermal and catalytic cracking of heavy petroleum fractions. Although propylene and propane have close boiling points, they are traditionally separated by distillation. Because distillation requires a large number of stages and considerable reflux and boil up flow rates compared to the feed flow, considerable attention has been given to the possible replacement of distillation with a more economical and less energy-intensive option. Based on the given properties of both species, propose some alternative properties that can be exploited to enhance the selectivity of propylene and propane separation. What kind of separation processes are based on these alternative properties?

Property

Propylene

Propane

Molecular weight Vd Waals volume (m/kmol) Vd Waals area (m/kmol) Acentric factor Dipole moment (debye) Radius of gyration (m*) Melting point (K) Boiling point (K) Critical temperature (K) Critical pressure (MPa)

. . . . . . . . . .

. . . . . . . . . .

Chapter 2 Evaporation and Distillation 2.1 Separation by evaporation 2.1.1 Introduction A large part of the separations of individual substances in a homogeneous liquid mixture or complete fractionation of such mixtures into their individual pure components is achieved through evaporative separations. Evaporative separations are based on the difference in composition between a liquid mixture and the vapor formed from it. This composition difference arises from differing effective vapor pressures, or volatilities, of the components in the liquid mixture. The required vapor phase is created by partial evaporation of the liquid feed through adding heat, followed by total condensation of the vapor. Due to the difference in volatility of the components, the feed mixture is separated into two or more products whose compositions differ from that of the feed. The resulting condensate is enriched in the more volatile components, in accordance with the vapor–liquid equilibrium (VLE) for the system at hand. When a difference in volatility does not exist, separation by simple evaporation is not possible. The basis for planning evaporative separations is knowledge of the VLE. Technically, evaporative separations are the most mature separation operations. Design and operating procedures are well established. Only when VLE or other data are uncertain, a laboratory and/or pilot plant study is necessary prior to the design of a commercial unit. The most elementary form is simple distillation in which the liquid mixture is brought to boiling, partially evaporated and the vapor formed is separated and condensed to form a product. This technique is commonly used in the laboratory for the recovery and/or purification of products after synthesis in an experimental setup as illustrated in Figure 2.1.

Temperature indicators

Cooler

Liquid mixture Heater

Distillate

Figure 2.1: Laboratory distillation setup. https://doi.org/10.1515/9783110654806-002

18

Chapter 2 Evaporation and Distillation

2.1.2 Vapor–liquid equilibria It is clear that equilibrium distributions of mixture components in the vapor and liquid phases must be different if separation is to be made by evaporation. At thermodynamic equilibrium, the compositions are called VLE that may be depicted as in Figure 2.2. For binary mixtures, the effect of distribution of mixture components between the vapor and liquid phases on the thermodynamic properties is illustrated in Figure 2.3. Figure 2.3a shows a representative boiling point diagram with equilibrium compositions as functions of temperature at a constant pressure. Commonly, the more volatile (low boiling) component is used to plot the liquid and vapor compositions of the mixture. The lower line is the liquid bubble point line, the locus of points at which a liquid on heating forms the first bubble of vapor. The upper line is the vapor dew point line, representing points at which a vapor on cooling forms the first drop of condensed liquid. The region between the bubble and dew point lines is the two-phase region, where vapor and liquid coexist in equilibrium. At the equilibrium temperature Teq, the liquid with composition xeq is in equilibrium with vapor composition yeq. Figure 2.3b displays a typical y–x diagram that is obtained by plotting the vapor composition that is in equilibrium with the liquid composition at a fixed pressure or a fixed temperature. At equilibrium, the concentration of any component i present in the liquid mixture is related to its concentration in the vapor phase by the equilibrium ratio as defined in eq. (1.2), also called the distribution coefficient Ki: Ki ≡

yi xi

(2:1)

where yi is the mole fraction of component i in the vapor phase and xi the mole fraction of component i in the liquid phase. The distribution coefficient is a function of

Figure 2.2: Schematic vapor–liquid equilibrium.

2.1 Separation by evaporation

(a)

19

(b) 1

Ptot = constant

T1

Vapor

Ptot, Teq yeq

Dew point line Temperature

Teq

y Equilibrium line

yeq

xeq

x = y line

Vapor and liquid Bubble point line

T2

Liquid

xeq 0

0

x,y

0

1

x

1

Figure 2.3: Isobaric vapor–liquid equilibrium diagrams: (a) dew and bubble point and (b) y–x diagram.

temperature and pressure only, not of compositions. The more volatile components in a mixture will have the higher values of Ki, whereas less volatile components will have lower values of Ki. The key separation factor in distillation is the selectivity of a component relative to a reference component. This selectivity in VLE usually referred to as relative volatility is defined in eq. (1.3) as follows: αij =

Ki yi =xi = Kj yj =xj

(2:2)

where component i is the more volatile one. The higher the value of the relative volatility, the more easily components may be separated by distillation. In ideal systems, the behavior of vapor and liquid mixtures obeys Dalton’s and Raoult’s laws. Dalton’s law relates the concentration of a component present in an ideal gas or vapor mixture to its partial pressure: pi = yi Ptot

(2:3)

where pi is the partial pressure of component i in the vapor mixture and Ptot is the total pressure of the system given by the sum of the partial pressures of all components in the system: Ptot =

N X i=1

pi

(2:4)

20

Chapter 2 Evaporation and Distillation

Raoult’s law relates the partial pressure of a component in the vapor phase to its concentration in the liquid phase, xi: pi = xi Pio

(2:5)

where Pio is the vapor pressure of pure component i at the system temperature (saturation pressure). Combining eqs. (2.3) and (2.5) yields: yi Ptot = xi Pio

(2:6)

This results in the following relations between the pure component vapor pressure, Pio, its distribution coefficient, Ki, and the relative volatility of two components in an ideal mixture, αij: Ki =

yi Pio = xi Ptot

and

αij =

Ki Pio = Kj Pjo

(2:7)

The latter equation shows that for ideal systems, the relative volatility is independent of pressure and composition. For a binary system with components A and B, where yA = 1 – yB and xA = 1 – xB, the relative volatility equation, eq. (2.2), can be rearranged to give y x =α· 1−y 1−x

or

y=

αx 1 + ðα − 1Þ x

(2:8)

where x and y represent the mole fractions of the more volatile component A. This equation is used to express the concentration of a component in the vapor as a function of its concentration in the liquid and relative volatility. This function is plotted in Figure 2.4 for various values of relative volatility. When relative volatility increases, the concentration of the most volatile component in the vapor increases. When the relative volatility is equal to 1, the concentrations of the most volatile component in the liquid and vapor phases are equal and a vapor/liquid separation is not feasible. At very low concentrations, eq. (2.8) reduces to a linear relation,1 represented as tangents in Figure 2.4: y=a·x

(2:9)

Since vapor pressures of components depend on temperature, equilibrium ratios are a function of temperature. However, the relative volatility considerably less sensitive to temperature changes because it is proportional to the ratio of the vapor pressures. In general, the vapor pressure of the more volatile component tends to increase at a

1 See also page 25.

2.1 Separation by evaporation

21

1 α=4 yeq (α = 4)

α = 2.5

yeq (α = 2.5) y

α=1

yeq = xeq

xeq

0 0

x

1

Figure 2.4: Vapor–liquid equilibrium compositions as a function of relative volatility.

slower rate with increasing temperature than the less volatile component. Therefore, the relative volatility generally decreases with increasing temperature and increases with decreasing temperature. Most liquid mixtures are nonideal and require Raoult’s law to be modified by including a correction factor called the liquid phase activity coefficient: pi = γi xi Pio

(2:10)

Although at high pressures, the vapor phase may also depart ideal vapor mixture, the common approach in distillation is to assume ideal vapor behavior and to correct nonideal liquid behavior with the liquid phase activity coefficient. The standard state for reference for the liquid phase activity coefficient is commonly chosen as γI ! 1 for the pure component. The liquid phase activity coefficient is strongly dependent upon the composition of the mixture. Positive deviations from ideality ( γI > 1) are more common when the molecules of different compounds are dissimilar and exhibit repulsive forces. Negative deviations ( γI < 1) occur when there are attractive forces between different compounds that do not occur for either component alone. For nonideal systems the relations for the distribution coefficient and relative volatility as given in eq. (2.7) now become: Ki =

γi Pio , Ptot

αij =

Ki γi Pio = Kj γj Pjo

(2:11)

In nonideal systems, the distribution coefficients and relative volatility are dependent on composition because of the composition dependence of the activity coefficients. When the activity coefficient of a specific component becomes high enough, an azeotrope may be encountered, meaning that the vapor and liquid compositions are equal

22

Chapter 2 Evaporation and Distillation

and the components cannot be separated by conventional distillation. Figure 2.5 shows binary vapor–liquid composition (x–y), temperature–composition (T–x) and pressure–composition (P–x) diagrams for nonazeotrope, minimum azeotrope and maximum azeotrope systems. A minimum boiling azeotrope boils at a lower temperature than each of the components in their pure states. When separating the components of this type of system by distillation, such as ethanol–water, the overhead product is the azeotrope. A maximum boiling azeotrope boils higher than

I. Intermediate boiling systems, benzene – toluene Ptot = const T2

y1

T = const p1

Vapor Liquid V+L V+L

Liquid T1

Vapor p2

(a)

x1,y1

(b)

x1,y1

(c)

x1

II. Minimum boiling azeotrope systems, ethanol – water Ptot = const

y1

T = const

T2

Liquid Vapor

p1

V+L T1

V+L

Vapor Liquid p2

(a)

xAZ

xAZ

xAZ (b)

x1,y1

x1,y1

(c)

x1

III. Maximum boiling azeotrope systems, acetone – chloroform Ptot = const

y1

T = const

p1

Vapor T2

V+L

Liquid V+L

p2

Liquid T1

Vapor

xAZ (a)

xAZ

xAZ x1,y1

(b)

x1,y1

(c)

x1

Figure 2.5: Types of binary temperature–composition (a), pressure–composition (b) and x–y phase diagrams for vapor–liquid equilibrium (c).

2.2 Separation by single-stage partial evaporation

23

either component in their pure states and is the bottom product of distillation. An example of this type of system is acetone–chloroform. Vapor pressures for many compounds are published in literature (see page 50) and often correlated as a function of temperature by the Antoine equation: ln Pio = Ai −

Bi T + Ci

(2:12)

where Ai, Bi and Ci are Antoine constants of component i, and T represents the temperature. Unfortunately, in some reference books, these data are given in non-SI units such as pressure in mmHg and, sometimes, temperature in °F. Then appropriate conversions are in order.

2.2 Separation by single-stage partial evaporation 2.2.1 Differential distillation Two main modes are utilized for single-stage separation by partial evaporation. The most elementary form is differential distillation, in which a liquid is charged to the still pot and heated to boiling. As illustrated in Figure 2.6, the method has a strong resemblance with laboratory distillation, shown previously in Figure 2.1. The vapor formed is continuously removed and condensed to produce a distillate.

Condenser Vapor line

Distillate receiver Charge

Stillpot

Heating medium

Figure 2.6: Simple differential distillation.

24

Chapter 2 Evaporation and Distillation

Usually the vapor leaving the still pot with composition yD is assumed to be in equilibrium with perfectly mixed liquid in the still at any instant. The distillate is richer in the more volatile components and the residual unvaporized bottoms are richer in the less volatile components. As the distillation proceeds, the relative amount of volatile components composition of the initial charge and distillate decrease with time. Because the produced vapor is totally condensed, yD = xD, there is only one single equilibrium stage, the still pot. Simple differential distillation is not widely used in industry, except for the processing of high-valued chemicals in small production quantities or for distillations requiring regular sanitation.

2.2.2 Flash distillation Flash distillation is the continuous form of simple single-stage equilibrium distillation. In a flash, a continuous feed is partially vaporized to give a vapor richer in the more volatile components than the remaining liquid. Because of the single stage, usually only a limited degree of separation is achieved. However, in some cases, such as the seawater desalination, the large volatility difference between the components results in complete separation. A schematic diagram of the equipment for flash distillation is shown in Figure 2.7. The pressurized liquid feed is heated and either flashed adiabatically across a nozzle to a lower pressure or, without the valve, partially vaporized isothermally. Either way, the created vapor phase is separated from the remaining liquid in a flash drum. A demister or

Vapor (V, yi) Ptot, TV = TL

Demister

Feed (TF, F, zi)

Q

Vapor Liquid

Liquid (L, xi) Figure 2.7: Flash drum: adiabatic flash distillation with valve or isothermal flash without valve.

2.2 Separation by single-stage partial evaporation

25

entrainment eliminator is often employed to prevent liquid droplets from being entrained in the vapor. When properly designed, the system in the flash chamber behaves very close to an equilibrium stage (TV = TL) due to the intimate contact between liquid and vapor. The designer of a flash system needs to know the liquid and vapor compositions that correspond to the operating pressure and temperature in the flash drum. The design calculations2 are based on the mass balances for the balance envelope shown as a dashed line in Figure 2.7 and on VLE data; x, y and z represent the compositions (mole fractions) of the liquid, vapor and feed, respectively. In a binary system, the two independent mass balances are the overall mass balance and the component balance for the more volatile component: F = V + L and

Fz = Vy + Lx

(2:13)

By introducing the liquid vapor ratio L/V, the latter two equations reduce to the dimensionless operating line equation:   L L z (2:14) y= − x+ 1+ V V Equilibrium data may be represented by relations such as given in eq. (2.1), which take the form in case of a binary system with components A and B: y = KA x

and

1 − y = KB ð1 − xÞ

(2:15)

By combining these two equilibrium line may be expressed equally well in terms of relative volatility: y=

αx with α = KA =KB 1 + ðα − 1Þ x

(2:16)

In case of ideal gas and liquid phases, the relative volatility α is the VL-equilibrium constant, also referred to as K (see, e.g., eq. (3.21) and (4.5)). In case of an isothermal flash no valve is used while the specified temperature is set through the heat exchanger. Gibbs’ phase rule3 states that in a binary two-phase system, two degrees of freedom exist; thus, one more parameter can be chosen freely. Given the feed composition, z, one option to define the system would be to specify temperature and liquid–vapor ratio, another option being the specification of temperature and total pressure. In the following two sections, we will show the choice of the relevant set of equations and how to solve these.

2 Estimation of the drum size related to the liquid and vapor fluxes is outside the scope of this textbook. 3 Degrees of freedom = number of components – number of phases + 2.

26

Chapter 2 Evaporation and Distillation

2.2.2.1 Specifying T and L/V In this case, the total pressure is not known beforehand and, noting that KA and KB are inversely proportional to Ptot, eq. (2.16) should be used as equilibrium line. Vapor and liquid compositions follow from solving eqs. (2.8) and (2.14) with known values of α and L/V. The partial pressures are calculated from eq. (2.5) and the total pressure from eq. (2.4). A graphical solution is very convenient since the simultaneous solution of eqs. (2.16) and (2.14) is represented by the intersection of the equilibrium curve and the operating line. Instead of the vapor–liquid ratio, often the fraction feed remaining liquid q and fraction feed vaporized 1 – q are utilized: q≡

L F

and

ð1 − qÞ =

V F

(2:17)

The operating line equation now becomes y = −

q 1 x + z 1−q 1−q

(2:18)

The x–y diagram in Figure 2.8 shows an example for a benzene–water separation, three straight operating lines and the equilibrium line4 according to eq. (2.16). Each operating line represents a fraction of feed evaporated. The intersection points give the vapor and liquid compositions, yeq and xeq, leaving the flash drum as the fraction of feed evaporated (1 – q) varies from 0.001 to 0.002 to 0.005. 2.2.2.2 Specifying T and Ptot Setting temperature and total pressure defines the liquid–vapor system in the flash drum. This is shown in Figure 2.3a, where at a given T and Ptot the compositions of the coexisting phases, xeq and yeq, are fixed. In this case, q (hence the slope of the operating line) is unknown beforehand; hence, the solution procedure now differs from that outlined in the previous section. However, three boundary conditions can be defined: – the feed composition, z, must fall between xeq and yeq to assure that a VLE exists at the given temperature and pressure, – in a vapor–liquid system q is be bounded between zero and unity, and – the condition that the sum of mole fractions in each phase equals unity, but in the form Σxi = Σyi . Now, the usual numerical procedure is to first find q by elimination of x and y from the operating line (eq. (2.14)) and equilibrium line equation (eq. (2.15)), noting that at given T and Ptot the distribution coefficients Ki are known constants.

4 Almost a straight line because of the very low values of the benzene mole fraction in the mixture.

27

2.2 Separation by single-stage partial evaporation

0.40

Benzene Mole Fraction Vapor

Equilibrium line

(1–q) = 0.001

0.20 (1–q) = 0.002

Operating lines

(1–q) = 0.005

yeq

xeq 0 0

0.20 0.40 0.60 Benzene Mole Fraction Liquid (× 1,000)

Figure 2.8: Graphical determination of the equilibrium benzene mole fraction in the vapor and liquid from a flash drum for several (1 – q) values.

In a multicomponent mixture yi = Ki·xi and the operating line equation becomes xi Ki = −

q zi xi + 1−q 1−q

or

xi =

zi Ki + ð1 − Ki Þq

(2:19)

But yi = Ki xi =

Ki zi Ki + ð1 − Ki Þq

(2:20)

and combination of eqs. (2.19) and (2.20) with the boundary condition Σxi = Σyi leads to X zi ð1 − Ki Þ =0 Ki + ð1 − Ki Þq i

(2:21)

This procedure for a multicomponent mixture thus leads to a single equation, which can be solved for the one unknown q. Once q is found between zero and unity, all xi and yi are calculated from eqs. (2.19) and (2.20).

28

Chapter 2 Evaporation and Distillation

Solving eq. (2.21) for q in a binary system (zA = z and zB = 1 – z) and substitution of the result in eq. (2.19) gives q= −z

KB KA − ð1 − zÞ 1 − KB 1 − KA

and

x=

1 − KB KA − KB

(2:22)

Note that KA and KB are functions of temperature and total pressure. This expression for x can also be obtained by elimination of y and α from eqs. (2.6), (2.7) and (2.8).

2.3 Multistage distillation 2.3.1 Distillation cascades Flash distillation is a very simple unit operation that in most cases produces only a limited amount of separation. Increased separation is possible in a cascade of flash separators that produce one pure vapor and one pure liquid product. Within the cascade, the intermediate product streams are used as additional feeds. The liquid streams are returned to the previous flash drum, while the produced vapor streams are forwarded to the next flash chamber. Figure 2.9 shows the resulting countercurrent cascade, because vapor and liquid streams go in opposite directions. The advantages of this cascade are that there are no intermediate products and the two end products can both be pure and obtained in high yield.

Distillate Distillate

Distillate

Feed

Feed

Feed

Residue

Residue

Figure 2.9: Flash drum cascades.

Residue

2.3 Multistage distillation

29

Although a significant advance, this multiple flash drum system is seldom used industrially. Operation and design is easier if part of the top vapor stream is condensed and returned to the first stage, reflux, and if part of the bottom liquid stream is evaporated and returned to the bottom stage, boil up. This allows control of the internal liquid and vapor flow rates at any desired level by applying all of the heat required for the distillation to the bottom reboiler and do all the required cooling in the top condenser. Partial condensation of intermediate vapor streams and partial vaporization of liquid streams is achieved by heat exchange between all pairs of passing streams. This is most effectively achieved by building the entire system in a column instead of the series of individual stages shown in Figure 2.9. Intermediate heat exchange is the most efficient with the liquid and vapor in direct contact. The final result is a much simpler and cheaper device, the distillation column shown in Figure 2.10. Total condenser

Reflux drum

Tray 1 Tray 2

Distillate

Reflux

Vapor Feed Liquid

Tray N Partial reboiler Bottoms

Figure 2.10: General flowchart of a plate distillation column.

2.3.2 Column distillation Most commercial distillations involve some form of multiple staging in order to obtain better separation than is possible by single vaporization and condensation. It is the most widely used industrial method of separating liquid mixtures in the chemical

30

Chapter 2 Evaporation and Distillation

process industry. As shown by Figure 2.10, most multistage distillations are continuously operated column-type processes separating components of a liquid mixture according to their different boiling points in a more volatile distillate and a less volatile bottoms or residue. The feed enters the column at the equilibrium feed stage. Vapor and liquid phases flow countercurrently within the mass transfer zone of the column where trays or packings are used to maximize interfacial contact between the phases. The section of column above the feed is called the rectification section and the section below the feed is referred to as the stripping section. Although Figure 2.10 shows a distillation column equipped with only 12 sieve trays, industrial columns may contain more than 100. The liquid from a tray flows through a downcomer to the tray below and vapor flows upward through the holes in the sieve tray to the tray above. Intimate contact between the vapor and liquid phases is created as the vapor passes through the holes in the sieve tray and bubbles through the pool of liquid residing on the tray. The vapors moving up the column from equilibrium stage to equilibrium stage are increasingly enriched in the more volatile components. Similarly, the concentration of the least volatile components increases in the liquid from each tray going downward. The overhead vapor from the column is condensed to obtain a distillate product. The liquid distillate from the condenser is divided into two streams. Part of it is withdrawn as overhead product (D) while the remaining distillate is refluxed (Lo) to the top tray to enrich the vapors. The reflux ratio is defined as the ratio of the reflux rate to the rate of product removal. The required heat for evaporation is added at the base of the column in a reboiler, where the bottom tray liquid is heated and partially vaporized to provide the vapor for the stripping section. Plant-size distillation columns usually employ external steam-powered kettle or vertical thermosyphon-type heat exchangers (Figure 2.11). Both can provide the amount of heat transfer surface required for large installations. Thermosyphon reboilers are favored when the bottom product

V ’’

V ’’

Steam

Steam

Bottoms B

L’’ Bottoms B

L’’ (a)

(b)

Figure 2.11: Industrial reboilers: (a) kettle type and (b) vertical thermosyphon type.

2.3 Multistage distillation

31

contains thermally sensitive compounds, only a small temperature difference is available for heat transfer and heavy fouling occurs. The vapor from the reboiler is sent back to the bottom tray and the remaining liquid is removed as the bottom product.

2.3.3 Feasible distillation conditions Industrial distillation processes are restricted by operability of the units, economic conditions and environmental constraints. An upper limit exists for feasible operating temperatures. One reason is the thermal stability of the species in the mixtures to be separated. Many substances decompose at higher temperatures and some species are not even stable at their normal boiling points. A second reason for a maximum temperature limitation is the means of heat supply. In most cases, the required energy is supplied by condensing steam. The pressure of the available steam places an upper limit on temperature levels that can be achieved. Only in special cases (crude oil, sulfuric acid), higher temperatures may be realized by heating with hot oil or natural gas burners. With high-pressure steam, the maximum attainable temperatures are limited to 300 °C, which can be raised to temperatures as high as 400 °C when other media are used. The temperature of the coolant for the overhead condenser dictates the lower limit of feasible temperatures in distillation columns. In most cases, water is used, resulting in a minimum temperature in the column of 40–50 °C. In most cases, temperature constraints can be met by selecting a proper operating pressure. Operating temperatures can be decreased by the use of a vacuum. It is technically feasible to operate distillation columns at pressures up to 2 mbar. Lower pressures are seldom used because of the high operating and capital costs of the vacuum-producing equipment. A column can be operated at higher pressure to increase the boiling point of low boiling mixtures. The upper limit for the operating pressure lies in the range of the critical pressures of the constituents. In addition to temperature and pressure, the feasible number of theoretical stages of industrial columns is also limited. Only in exceptional cases, commercial distillation columns are constructed that contain more than 100 stages.

2.3.4 External column balances Once the separation problem has been specified, the task of the engineer is to calculate the design variables. The first step is the development of mass balances around the entire column using the balance envelope shown by the dashed outline in Figure 2.12. The aim of these balances is to calculate flow rates and compositions for the distillate and bottom products, D and B, for the given problem specification. Analogously to the flash calculations in Section 2.2.2, the overall mass balance becomes

32

Chapter 2 Evaporation and Distillation

Condenser

Reflux 1 Distillate 1

Feed F, xF

D, xD

Rectifying section stages

f

Feed stage

Stripping section stages N

Reboiler Bottoms B, xB

Figure 2.12: Distillation column overall mass balance envelope.

F = D+B

(2:23)

In binary distillation, the specification of xD and xB for the more volatile component fixes the distillate and bottoms flow rates D and B through the additional overall component material balance: xF F = xD D + xB B

(2:24)

2.4 McCabe–Thiele analysis 2.4.1 Internal balances If the molar heat of vaporization for any component has approximately the same value, condensation of 1 mol of vapor will vaporize 1 mol of liquid. Thus, liquid and vapor flow rates tend to remain approximately constant as long as heat losses from the column are negligible and the pressure is uniform throughout the column. This simplified situation is closely approximated for many distillations; constant molar vapor and liquid flow may be assumed in each section of the distillation column,

2.4 McCabe–Thiele analysis

33

given a value for the distillate flow D. For constant molar flow conditions, it is not necessary to consider energy balances in either the rectifying or stripping sections. Only material balances and a VLE curve are required. Based on these assumptions, the McCabe–Thiele method provides a graphical solution to the material balances and equilibrium relationships inside the column. In this chapter, the graphical McCabe–Thiele method is used to visualize the essential aspects of multistage distillation calculations required to understand the fundamentals behind modern, computer-aided design methods. In addition to the number of equilibrium stages N, also the required minimum number of stages Nmin, minimal reflux ratio Rmin and optimal stage for feed entry are derived. Finally, the overall energy balances are derived to determine condenser and reboiler duties. 2.4.1.1 Rectifying section As illustrated in Figure 2.13, the rectifying section extends from the top stage 1 to just above the feed stage f. For the indicated balance envelope, the material balance for the more volatile component over the total condenser and a top portion of rectifying stages 1 to n is written as V ′n + 1 yn + 1 = L′n xn + D xD

(a)

(2:25)

(b)

Figure 2.13: (a) Mass balance envelope and (b) operating line for the rectifying section.

Since V′ and L′ are constant throughout the rectifying section, eq. (2.25) can be rewritten as: yn + 1

    L′ D xn + xD = V′ V′

(2:26)

34

Chapter 2 Evaporation and Distillation

Equation (2.22) is the operating line in the rectifying section. It relates the concentrations yn+1 and xn of the two passing streams V′ and L′ in the column, and thus represents the mass balances in the rectifying section. The slope of the operating line is L′/V′, which is constant and L′. The liquid entering the top stage is the external reflux rate, L′, and its ratio to the distillate rate, L′/D, is defined as the reflux ratio R. Since V′ = L′ + D, the slope of the operating line is readily related to the reflux ratio: L′ L′ R = = R+1 V′ L′ + D

and

D D 1 = = R+1 V′ D + L′

(2:27)

Substituting into eq. (2.26) produces the most useful form of the operating line for the rectifying section:     R 1 xn + xD yn + 1 = (2:28) R+1 R+1 As shown in Figure 2.13, the operating line plots as a straight line in the yx-diagram with an intersection at y = xD on the y = x line, for specified values of R and xD. 2.4.1.2 Stripping section The stripping section extends from the feed to the bottom stage. Analogous to the rectifying section, a bottom portion of the stripping stages, including the partial reboiler and extending up from stage N to stage m + 1, located somewhere below the feed is considered. A material balance for the more volatile component over the envelope indicated in Figure 2.14 results in ′′ xm = V ′′m + 1 ym + 1 + B xB Lm

Figure 2.14: (a) Mass balance envelope and (b) operating line for the stripping section.

(2:29)

2.4 McCabe–Thiele analysis

35

where the constant-molar-flow conditions rearrange into the operating line representing the mass balance of the passing streams V″ and L″ in the stripping section:     L′′ B xm − xB ym + 1 = (2:30) ′′ V V ′′ The slope of this operating line for the stripping section is seen to be >1 because L″ > V″. This is the reverse of the conditions in the rectifying section. For known values of L″, V″ and xB, eq. (2.30) can be plotted as a straight line with an intersection at y = xB on the y = x line and a slope of L″/V″, as shown in Figure 2.14. 2.4.1.3 Feed stage considerations Thus far we have not considered the feed to the column. In the rectifying and stripping sections of the column, the operating line represents the mass balances. When material is added or withdrawn from the column, the mass balances will change and the operating lines will have different slopes and intercepts. The addition of the feed increases the liquid flow in the stripping section, increases the vapor flow in the rectifying section, or does both. In the rectifying section, the vapor flow is greater than the liquid flow rate. In the stripping section, the liquid flow is greater than the vapor flow rate. Obviously, the phase and temperature of the feed affect the vapor and liquid flow rates in the column. For instance, if the feed is liquid, the liquid flow rate below the feed stage must be greater than the liquid flow above the feed stage, L″ > L′. Similarly V′ should be larger than V″ if the feed is a vapor. These effects can be quantified by writing mass and energy balances around the feed stage shown schematically in Figure 2.15a. The overall mass balance for the balance envelope is F + V ′′ + L′ = L′′ + V ′

Figure 2.15: Mass balance envelope (a) and q-line (b) for the feed stage.

(2:31)

36

Chapter 2 Evaporation and Distillation

and the related balance over just the liquid phase reads: LF = L′′ − L′ ≡ q · F

(2:32)

where q is defined as the liquid fraction in the feed at feed stage temperature, similar to the q-factor defined in Section 2.2.2. Under the condition that the feed is at feed stage temperature, q represents the moles of the liquid flow in the stripping section that result from the introduction of each mole of feed. Assuming that the molar vapor enthalpies hV nor the molar liquid enthalpies hL vary much with composition, the energy balance across the feed stage equals F hF + V ′′ hV + L′ hL = L′′ hL + V ′ hV

(2:33)

Rewriting the mass balance equation (2.31) in terms of V″–V′ and substituting into the energy balance gives F hF + ðL′′ − L′Þ hV − F hV = ðL′′ − L′Þ hL

(2:34)

which is easily rearranged to yield q ≡

L′′ − L′ h V − hF = F h V − hL

(2:35)

Hence, the liquid feed fraction equals the ratio of the heat required to completely evaporate a mole of feed hV ‒ hF and the molar heat of evaporation hV ‒ hL. Since the liquid and vapor enthalpies can be estimated, we can easily calculate q from eq. (2.35) for the condition that the feed consists of a partially evaporated liquid and vapor mixture (1 > q > 0). The feed may also be a saturated liquid at its bubble point (q = 1), or a saturated vapor at its dew point (q = 0). The energy balance in eq. (2.33) does not comprise a term to bring the feed to the feed stage temperature. Thus, if the feed is introduced as cold liquid (q > 1) or as a superheated vapor (q < 0), the value of q calculated from eq. (2.35) should be corrected for the heat involved.5 The required correction is outside the scope of this textbook but can be found in the literature references. 2.4.1.4 Feed line After quantifying the value for q, the changes in liquid and vapor flow rates due to the introduction of the feed on the feed stage are calculated. The contribution of the feed stream to the internal flow of liquid is q·F (eq. (2.32)), so the total flow rate of liquid and vapor in the stripping section becomes

5 We will denote q-values subject to this temperature correction as q′. For small temperature differences, q′ ≈ q.

2.4 McCabe–Thiele analysis

L′′ = L′ + q F

and

V ′′ = V ′ − ð1 − qÞ F

37

(2:36)

Although with these values for L″ and V″ in eq. (2.30) can be used to locate the stripping operating line, it is more common to use an alternative method that involves the feed-line or q-line. The intersection point of the rectifying and stripping operating lines is derived by writing the mass balance equations for both sections for constant molar flow: V ′ y = L′ x + D xD

and

V ′′ y = L′′ x − B xB

(2:37)

Subtracting both equations gives y ðV ′ − V ′′Þ = x ðL′ − L′′Þ + D xD + B xB = x ðL′ − L′′Þ + F xF

(2:38)

which can be rearranged into y =

L′′ − L′ F x − xF V ′′ − V ′ V ′′ − V ′

(2:39)

Substituting L″–L′ and V″–V′ from eq. (2.36) and some rearrangements results in the equation for the feed-line or q-line: y = −

q 1 x + xF 1−q 1−q

(2:40)

This equation represents a straight line of slope –q/(1–q) passing through the point (yF,xF) on which every possible intersection point of the two operating lines for a given feed must fall. The position of the line depends only on xF and q′. Thus, by changing the reflux ratio, the point of intersection is changed but remains on the q-line. From the definition of q′, it follows that the slope of the q-line is governed by the nature of the feed. Examples of the various types of feeds and the slopes of the q-line are illustrated in Figure 2.16. Note that all the feed lines intersect at one point, which is at y = x = xF. Following the placement of the rectifying section operating line and the q-line, the stripping section operating line is located by drawing a straight line from the point (y = xB, x = xB) on the y = x line through the point of intersection with the q-line and the rectifying section operating line as shown in Figure 2.15.

2.4.2 Required number of equilibrium stages 2.4.2.1 Graphical determination of stages and location of feed stage Once the equilibrium line, operating lines and q-line have been plotted, we can continue with the determination of the number of equilibrium stages required for the entire column and the location of the feed stage. The operating lines relate the solute concentration in the vapor passing upward between two stages to the solute

38

Chapter 2 Evaporation and Distillation

Figure 2.16: Dependence of q-line on feed condition (for q′ see footnote 5 on page 36).

concentration in the liquid passing downward between the same two stages. The equilibrium curve relates the solute concentration in the vapor leaving an equilibrium stage to the solute concentration in the liquid leaving the same stage. This makes it possible to determine the required number of stages and the location of the feed stage by constructing a staircase between the operating line and the equilibrium curve, as shown in Figure 2.17. Although the construction can start either from the top, the bottom or from the feed stage, it is common that the staircase is stepped of from the top and continued all the way to the bottom. Starting from the point (y1= xD, x0= xD) on the rectifying section operating line and the y = x line, a horizontal line is drawn to the left until it intersects the equilibrium curve at (y1,x1), the compositions of the equilibrium phases leaving the top equilibrium stage. A vertical line is now dropped until it intersects the operating line at the point (y2,x1), the compositions of the two phases passing each other between stages 1 and 2. The horizontal and vertical line construction is continued down the rectifying section until the feed stage is reached. At the feed stage, we switch to the stripping section operating line because of the different operating line in the stripping section due to the addition of feed. Although the separation shown in Figure 2.17 would require exactly six equilibrium contacts (five equilibrium stages plus an equilibrium partial reboiler), in practice hardly ever an integer number of stages result, but rather a fractional stage will appear at the top or at the bottom. Starting the staircase construction from the feed stage to both the top and the bottom stages, at either side generally a fraction of a theoretical

39

2.4 McCabe–Thiele analysis

Figure 2.17: McCabe–Thiele graphical determination of the number of equilibrium stages.

stage will be found (see Figure 2.18a). Either fraction of a theoretical stage can be determined from fraction =

distance from operating line to xB distance from operating line to equilibrium line

(2:41)

Figure 2.18: Determination of (a) minimum number of theoretical stages and (b) minimum reflux ratio in a McCabe–Thiele diagram.

40

Chapter 2 Evaporation and Distillation

In Figure 2.17, it is seen that the smallest number of total stages occurs when the transfer is made at the first opportunity after a horizontal line of the staircase crosses the q-line. This is the optimal feed stage location because the separation will require the fewest total number of stages when feed stage 3 is used. For binary distillation, the optimum feed stage will always be the stage where the step in the staircase includes the point of intersection of the two operating lines. 2.4.2.2 Limiting conditions For any distillation operation, there are infinite combinations of reflux ratios and numbers of theoretical stages possible. The larger the reflux ratio, the fewer theoretical stages are required but the more energy is consumed. For a given combination of feed, distillate and bottom compositions, there are two constraints that set the boundary conditions within which the reflux ratio and number of theoretical stages must be. The minimum number of theoretical stages, Nmin, and the minimum reflux ratio, Rmin. As the reflux ratio is increased, the slope of the rectifying section operating line increases to a limiting value of L′/V′ = 1, meaning that the system is at total reflux. Correspondingly, the slope of the stripping section operating line decreases to a limiting value of L″/V″ = 1. The minimum number of theoretical stages occurs when both operating lines coincide with the y = x line and neither the feed composition nor the q-line influences the staircase construction. An example of the McCabe–Thiele construction for this limiting condition is shown in Figure 2.18a. Because the operating lines are located as far away as possible from the equilibrium curve, a minimum number of stages is required. Column operation with minimum internal gas and liquid flow (i.e., minimum reflux) separates a mixture with the lowest energy input. As the reflux ratio decreases, the intersection of the two operating lines and the q-line moves from the y = x line toward the equilibrium curve. The number of equilibrium stages required increases because the operating lines move closer to the equilibrium curve. Finally, a limiting condition is reached when the point of intersection is on the equilibrium curve, as shown in Figure 2.18b. To reach that stage from either the rectifying section or the stripping section, an infinite number of stages would be required to achieve the desired separation. The corresponding minimum reflux ratio is easily determined from the slope of the limiting operating line (L′/V′)min for the rectifying section:       ðL′ V ′ min L′ L′    = = (2:42) Rmin = D min V ′ − L′ min 1 − L′ V ′ min

Both of these limits, the minimum number of stages and the minimum reflux ratio serve as valuable guidelines within which the practical distillation conditions must lie. The operating, fixed and total cost of a distillation system are a strong function

2.4 McCabe–Thiele analysis

41

of the relation of the operating reflux ratio to the minimum reflux ratio. As shown by Figure 2.19, first, the fixed cost decreases by increasing the reflux ratio because fewer stages are required but then rises again as the diameter of the column increases at higher vapor and liquid loads. Similarly, the operating cost for energy increases almost linearly as the operating reflux ratio increases. For most commercial operations, the optimal operating reflux ratios are in the range of 1.1–1.5 times the minimum reflux ratio.

Figure 2.19: Typical distillation fixed, operating and total costs as a function of reflux ratio.

2.4.2.3 Fenske and Underwood equations Alternative to the graphical determination, approximate values for the minimum number of stages and minimum reflux can also be obtained from the Fenske and Underwood equations, respectively. Both equations are based on the special situation that the relative volatility remains approximately constant throughout the column. The case of the minimum number of theoretical stages, Nmin, is achieved at total reflux, as discussed in the previous section. Total reflux exists when no product is withdrawn: D=B=0

(2:43)

42

Chapter 2 Evaporation and Distillation

hence V ′ − L′ = L′′ − V ′′ = 0

(2:44)

and the slope of both operating lines is unity; see eqs. (2.28) and (2.30). This means that the composition of the streams passing between two subsequent stages is equal: y2 = x1 , y3 = x2 , or, generally, yn + 1 = xn

(2:45)

For a binary equilibrium between component A and the less volatile component B on stage 1 (see Figure 2.13) eq. (2.1) applies: y1 = KA · x1 = KA · y2

(2:46)

y1 = KA2 · y3

(2:47)

For stage 2, it follows that

Similarly, repeating this procedure for a succession of N theoretical stages, the result becomes y1 = KAN · yN + 1

(2:48)

At the top of the column y1 = x0 = xD for a total condenser while for the total reboiler yN+1 = xN = xB. Hence, the connection between the mole fraction of A and B in the top and bottom products is xD = KAN · xB

and

1 − xD = KBN · ð1 − xB Þ

(2:49)

Finally, note that KA/KB = αAB, Nmin stages are needed to obtain the preset top and bottom compositions: xD xB = ðαAB ÞNmin 1 − xD 1 − xB

(2:50)

Solving the equation for Nmin by logarithms gives the Fenske equation that can be used to estimate the minimum number Nmin of required theoretical stages (Nmin – 1 plates plus the reboiler) in the column: h . i x xB ln 1 −Dx 1 − xB D Nmin = (2:51) ln αAB The Fenske equation is also applicable for multicomponent mixtures when the relative volatility is based on the light key relative to the heavy key. If the change in relative volatility from the bottom of the column to the top is moderate, a mean value of the relative volatility is generally calculated by taking the average of the relative volatility at the top, αD, and the bottom of the column, αB as follows:

2.4 McCabe–Thiele analysis

pffiffiffiffiffiffiffiffiffiffiffiffiffi αav = αB · αD

43

(2:52)

In a similar way, the minimum reflux ratio Rmin can be obtained analytically from the physical properties of a binary system. Starting again from a material balance (Figure 2.13), this time over the top portion of the column comprising n theoretical stages, gives: V ′ yn + 1 = L′ xn + D xD

(2:53)

V ′ ð1 − yn + 1 Þ = L′ ð1 − xn Þ + D ð1 − xD Þ

(2:54)

and

Under conditions of minimum reflux, a column has to have an infinite number of plates. For large values of n, approaching the pinch (see Figure 2.18b), the composition on plate n is equal to that on plate n + 1, so near the pinch yn + 1 = y∞ and xn = x∞

(2:55)

Application of the equilibrium relation equation (2.16), dividing eqs. (2.53) and (2.54) and introduction of the relative volatility αAB provide: y∞ αAB x∞ L′ x∞ + D xD . = = 1 − y∞ 1 − x∞ L′ ð1 − x∞ Þ + D ð1 − xD Þ

(2:56)

Solving for L′/D, this rearranges into the Underwood expression for estimating the minimum reflux ratio required to separate the liquid composition on a plate near the pinch into the desired distillate composition: Rmin =

  L′ D

=

xD x∞

min

− αAB

1 − xD 1 − x∞

αAB − 1

(2:57)

This expression applies to any feed condition q because the relation between the pinch and the feed composition, xF, is not defined yet. For q = 1, saturated liquid, x∞ = xF and the minimum reflux ratio for the desired separation of the feed composition becomes: Rmin =

xD xF

− αAB

1 − xD 1 − xF

αAB − 1

(2:58)

If the distillate product is required as a pure substance (xD = 1), this equation reduces to Rmin =

1 ðαAB − 1Þ xF

(2:59)

44

Chapter 2 Evaporation and Distillation

2.4.2.4 Use of Murphree efficiency The McCabe–Thiele method assumes that the two phases leaving each stage are in thermodynamic equilibrium. In industrial equipment, the equilibrium is usually not fully approached. The most commonly used stage efficiencies used to describe individual tray performance for individual components are the Murphree vapor and liquid efficiencies. The Murphree vapor efficiency EMV for stage n is defined as the ratio of the actual change in vapor composition over the change in vapor composition for an equilibrium stage: EMV =

yn − yn + 1 y*n − yn + 1

(2:60)

where y*n is the composition of the hypothetical vapor phase in equilibrium with the liquid composition leaving stage n. Once the Murphree efficiency is known for every stage, it can easily be used on a McCabe–Thiele diagram. The denominator represents the vertical distance from the operating line to the equilibrium line while the numerator is the vertical distance from the operating line to the actual outlet concentration. In stepping of stage, the Murphree vapor efficiency represents the fraction of the total vertical distance to move from the operating line to the equilibrium line. As only EMV of the total vertical path is traveled, we get the result shown in Figure 2.20.

Figure 2.20: Application of Murphree plate efficiency in the McCabe–Thiele construction.

2.5 Advanced distillation techniques

45

2.4.3 Energy requirements Following the determination of the feed condition, reflux ratio and number of theoretical stages, estimates of the heat duties of the condenser and reboiler can be made. When the column is well insulated, all of the heat transfer takes place in the condenser and reboiler, column pressure is constant, the feed is at the bubble point and a total condenser is used, the energy balance over the entire column gives Reboiler in = Condenser out

(2:61)

QR = Q C

(2:62)

or

The energy balance can by approximated by applying the assumptions of the McCabe–Thiele method, yielding for the reboiler and condenser duty: QR = QC = D ðR + 1Þ ΔHvap

(2:63)

If saturated steam is the heating medium for the reboiler, the required steam rate becomes Φm, steam =

QR · Msteam ΔHvap, steam

(2:64)

where ΔHvap, steam =Msteam is the specific enthalpy of vaporization of steam (≈2,100 kJ kg–1). The cooling water rate for the condenser is Φm, C =

QC CP, water ðTout − Tin Þ

(2:65)

where CP,water is the specific heat capacity of water (≈4.2 kJ kg–1 K–1). In general, the cost of cooling water can be neglected during a first evaluation because the annual cost of reboiler steam is an order of magnitude higher.

2.5 Advanced distillation techniques 2.5.1 Batch distillation In most of the large chemical plants, distillations are run continuously. For small production units, where most chemical processes are carried out in batches it is more convenient to distil each batch separately. In batch distillation, a liquid mixture is charged to a vessel where it is heated to the boiling point. When boiling begins, the vapor is passed through a fractionation column and condensed to obtain a distillate product, as indicated in Figure 2.21. As with continuous distillation, the

46

Chapter 2 Evaporation and Distillation

Figure 2.21: Schematic of a batch distillation unit.

purity of the top product depends on the still composition, the number of plates of the column and on the reflux ratio used. In contrast to continuous distillation, batch distillation is usually operated with a variable amount of reflux. Because the lower boiling components concentrate in the vapor and the remaining liquid gradually becomes richer in the heavier components, the purity of the top product will steadily drop. This is generally compensated by a gradual increase in reflux ratio during the distillation process to maintain a constant quality of the top product. To obtain the maximum recovery of a valuable component, the charge remaining in the still after the first distillation may be added to the next batch. The main advantage of batch distillation is that multiple liquid mixtures can be processed in a single unit. Different product requirements are easily taken into account by changing the reflux ratio. Even multicomponent mixtures can be separated into the different components by a single column when the fractions are collected separately. An additional advantage is that batch distillation can also handle sludge and solids. The main disadvantages of batch distillation are that for a given product rate, the equipment is larger and the mixture is exposed to higher temperature for a longer time. This increases the risk of thermal degradation or decomposition. Furthermore, it requires more operator attention, energy requirements are higher and its dynamic nature makes it more difficult to control and model.

2.5 Advanced distillation techniques

47

2.5.2 Continuous separation of multiple product mixtures Very often a separation unit incorporated in the process separates the stream into several products by means of several columns. Where the columns are located in the flow diagram will have a very great influence on the economics of the separation unit. Case studies have shown that operating costs alone may vary by a factor of 2 depending on the flow sheet chosen. If each component of a multicomponent distillation is to be essentially pure when recovered, the number of required columns is equal to the number of components minus one. Thus, a three-component mixture requires a two- and a four-component mixture requires three separate columns. Those columns can be arranged in many different ways as illustrated in Figure 2.22 for the two possible separation configurations of a ternary mixture.

Figure 2.22: Possible distillation configurations for separating ternary mixtures.

Note that it takes a sequence of two ordinary distillation columns to separate a mixture in three products. Because of the many possible flow sheet configurations for mixtures with more than three components it would be far too complex to find the best flow diagram by systematic study. Fortunately, some heuristic rules are available to provide guidelines for defining flow diagrams to accelerate the search for the optimal separation sequence of multicomponent mixtures: 1. Remove corrosive and hazardous materials first 2. First, eliminate majority components 3. Start out with easy separations 4. Sequences that remove the components one by one in column overheads should be favored 5. Give priority to separations that give a more nearly equimolal division of the feed between the distillate and bottoms product 6. End up with difficult separations such as azeotropic mixtures

48

Chapter 2 Evaporation and Distillation

7.

Separations involving very strict product specifications should be reserved until late in a sequence 8. Favor sequences that yield the minimum necessary number of products 9. Avoid vacuum distillation and refrigeration if possible

2.5.3 Separation of azeotropes When due to minimal difference in boiling point and/or highly nonideal liquid behavior, the relative volatility becomes lower than 1.1, ordinary distillation may be uneconomic and in case an azeotrope forms even impossible. In that event, enhanced distillation techniques should be explored. For these circumstances, the most often used technique is extractive distillation where a large amount of a relatively high boiling solvent is added to increase the relative volatility of the key components in the feed mixture. In order to maintain a high concentration throughout the column, the solvent is generally introduced above the feed entry and a few trays below the top. It leaves the bottom of the column with the less volatile product and is recovered in a second distillation column as shown in Figure 2.23. Because the high boiling solvent is easily recovered by distillation when selected in such a

Figure 2.23: Extractive distillation separation of toluene from methylcyclohexane (MCH) using high boiling polar solvents such as NMP, sulfolane, phenol and glycols.

2.5 Advanced distillation techniques

49

way that no new azeotropes are formed, extractive distillation is less complex and more widely used than azeotropic distillation. In homogeneous azeotropic distillation, an entrainer is added to the mixture that forms a homogeneous minimum or maximum boiling azeotrope with one or more feed components. The entrainer can be added everywhere in the column. If an entrainer is added to form an azeotrope, the azeotrope will exit the column as the overhead or bottom product leaving behind component(s) that may be recovered in the pure state. A classical example of azeotropic distillation is the recovery of anhydrous ethanol from aqueous solutions. Organic solvents such as benzene or cyclohexane are used to form desirable azeotropes, allowing the separation to be made. As shown in Figure 2.24, the ethanol azeotrope from the crude column overhead is fed to the azeotropic distillation column where cyclohexane is used to form an azeotrope with water. Ethanol is recovered as the bottom product. The overhead azeotropic cyclohexane/water mixture is condensed, where water-rich and cyclohexane-rich liquid phases are formed. Residual cyclohexane is removed from the water-rich phase in a stripper.

Figure 2.24: Dehydration of alcohol by azeotropic distillation with solvents such as benzene or cyclohexane as entrainer.

It is well known that many minimum boiling azeotrope-containing mixtures allow the position of the azeotrope to be shifted a change in system pressure. This effect can be exploited to separate a binary azeotrope containing mixture when appreciably changes (>5 mol%) in azeotropic composition can be achieved over a moderate

50

Chapter 2 Evaporation and Distillation

pressure range. Pressure swing distillation uses a sequence of two columns operated at different pressures for the separation of pressure-sensitive azeotropes. This is illustrated in Figure 2.25, where the effect of pressure on the temperature and composition of a minimum boiling azeotrope is given. The binary azeotrope can be crossed by first separating the component boiling higher than the azeotrope at low pressure. The composition of the overhead should be as close as possible to that of the azeotrope at this pressure. As the pressure is increased, the azeotropic composition moves toward a higher percentage of A and component B can be separated from the azeotrope as the bottom product in the second column. The overhead of the second column is returned to the first column.

Figure 2.25: Pressure swing distillation: T–y–x curves for minimum boiling azeotrope at pressures P1 and P2 (a) and distillation sequence for minimum boiling azeotrope (b).

Nomenclature B CP D DAB EMV F ΔHvap Ki L NOV

Bottom product flow Specific heat capacity Top product flow Coefficient of diffusion of A in a mixture of A and B Murphree or plate efficiency, eq. (.) Feed flow Molar heat of vaporization Distribution coefficient of component i Liquid flow (in rectifying section L′, in stripping section L″) Overall number of gas-phase transfer units

mol s– J kg– K– mol s– m s– – mol s– J mol– – mol s– –

Exercises

B N P P Ptot Q Q R T x, y, z V Φm αij Γ

Bottom product flow Number of (theoretical) stages Partial pressure Saturation pressure Total pressure Fraction-saturated liquid feed, eq. (.) Heat flow Reflux ratio L′/D, eq. (.) Temperature Mole fraction (liquid, vapor, feed) Vapor flow (in rectifying section V′, in stripping section V″) Mass flow rate Selectivity, relative volatility or equilibrium constant, eqs. (.), (.), (.), (.) Activity coefficient

51

mol s– – N m– N m– N m– – J s– – K – mol s– kg s– – –

Indices A,B,i,j B C D F L n, m R V

Components Bottom Condenser Distillate Feed Liquid Stage number Reboiler Vapor

Exercises 1

In Vapor-Liquid Equilibrium Data Collection, the following form of Antoine equation is used: log Pi0 = A′ −

B′ T + C′

with Pi0 = saturation pressure in mmHg and temperature T in °C (760 mmHg = 1 atm). For benzene in benzene–toluene mixtures, the following values are reported: A′ = 6.87987 ½ − , B′ = 1, 196.760 ½ −  and C′ = 219.161  C Calculate the constants A, B and C in the Antoine equation as defined in eq. (2.12): ln Pi0 = A −

B T+C

with pressure units in atm and temperature in °C.

52

2

Chapter 2 Evaporation and Distillation

With temperature in degrees Fahrenheit and pressure in pounds per sq. inch, the following Antoine constants apply for benzene: A′ = 5.1606, B′ = 2, 154.2, C′ = 362.49 ðpsi,  F,

10

log P expressionÞ

Note that 1 atm = 14.696 psi = 1.01325 bar and T (°F) = T (°C)·9/5 + 32, and calculate the saturation pressure of benzene in bar at 80.1 °C. 3

VLE data for benzene–toluene are given at 1 atm and at 1.5 atm in the following T–x diagram. Expressing the saturated vapor pressure in atm as a function of temperature in K, the Antoine constants for benzene and toluene are, respectively A = 9.2082, B = 2, 755.64, C = − 54.00 and  A = 9.3716, B = 3, 090.78, C = − 53.97 atm, K, eq.ð2.12ÞÞ Check the phase compositions at 100 °C and total pressures of 1.0 and 1.5 atm.

4

A liquid benzene–toluene mixture with z = 0.40 should produce a liquid with x = 0.35 in a flash drum at 1.0 bar. a. Calculate the required feed temperature. b. Calculate the equilibrium vapor–liquid ratio in the flash drum.

Exercises

53

5

We wish to flash distill isothermally a mixture containing 45 mol% of benzene and 55 mol% of toluene. Feed rate to the still is 700 mol/h. Equilibrium data for the benzene–toluene system can be approximated with a constant relative volatility of 2.5, where benzene is the more volatile component. Operation of the still is at 1 atm. a. Plot the y–x diagram for benzene–toluene. b. If 60% of the feed is evaporated, find the liquid and vapor compositions. c. If we desire a vapor composition of 60 mol%, what is the corresponding liquid composition and what are the liquid and vapor flow rates? d. Find the compositions and flow rates of all unknown streams for a twostage flash cascade where 40% of the feed is flashed in the first stage and the liquid product is sent to a second flash chamber where 30% is flashed.

6

Distillation is used to separate pentane from hexane. The feed amounts 100 mol/s and has a mole ratio of pentane/hexane = 0.5. The bottom and top products have the compositions xB= 0.05 and xD= 0.98. The reflux ratio is 2.25. The column pressure is 1 bar. The feed, at the bubbling point, enters the column exactly on the feed tray. The tray temperature is equal to the feed temperature. The pentane vapor pressure is given by    310 ðbarÞ ð1 atm = 1.013 barÞ po5 = exp 11* 1 − TðKÞ The vapor pressure of hexane is 1/3 of the pentane vapor pressure over the whole temperature range. The average density of liquid pentane and hexane amounts to 8,170, respectively, 7,280 mol/m3. The heat of vaporization amounts to 30 kJ/mol. The distance between the trays amounts to 0.50 m. a. Calculate the feed temperature. b. Calculate the vapor stream from the reboiler. c. Calculate the required energy in the reboiler. d. Construct the y–x diagram. e. Construct the operating lines and locate the feed line. f. Determine the number of equilibrium stages. g. Determine the height of the column.

7

Methanol (M) is to be separated from water (W) by atmospheric distillation. The feed contains 14.46 kg/h methanol and 10.44 kg/h water. The distillate is 99 mol% pure, while the bottom product contains 5 mol% of methanol. The feed is subcooled such that q = 1.12. a. Determine the minimum number of stages and minimum reflux. b. Determine the feed stage location and number of theoretical stages required for a reflux ratio of R = 1.

54

Chapter 2 Evaporation and Distillation

VLE data (1 atm, mole fraction methanol) x 0.0321 0.0523 0.075 0.154 0.225 y 0.1900 0.2940 0.352 0.516 0.593 8

0.349 0.703

0.813 0.918

0.918 0.963

A feed to a distillation unit consists of 50 mol% benzene in toluene. It is introduced to the column at its bubble point to the optimal plate. The column is to produce a distillate containing 95 mol% benzene and bottoms of 95 mol% toluene. For an operating pressure of 1 atm, calculate: a. the minimum reflux ratio; b. the minimum number of equilibrium stages to carry out the desired separation; c. the number of actual stages needed, using a reflux ratio (L′/D) of 50% more than the minimum; d. the product and residue stream in kilograms per hour of product if the feed is 907.3 kg/h; e. the saturated steam required in kilograms per hour for heat to the reboiler using the following enthalpy data: Steam: ΔHvap = 2,000 kJ/kg Benzene: ΔHvap = 380 kJ/kg Toluene: ΔHvap = 400 kJ/kg VLE data (1 atm, mole fraction benzene) x 0.10 0.20 0.30 0.40 0.50 0.60 y 0.21 0.37 0.51 0.64 0.72 0.79

0.70 0.86

0.80 0.91

0.90 0.96

9

During the synthesis of nitrotoluene, ortho-nitrotoluene as well as paranitrotoluene are formed. The customer requires pure para-nitrotoluene, the most volatile of the two isomers. The product requires a minimal purity of 95%. Another customer is interested in ortho-nitrotoluene, also with a minimal purity of 95%.The feed, of which 40% is in the vapor phase, amounts to 100 kmol/h and consists of 55% ortho-nitrotoluene. At the chosen operation pressure and temperature of the column, the relative volatility of ortho-nitrotoluene over para-nitrotoluene is 2.0. a. Calculate the size of the top and bottom streams. b. Determine the minimum number of stages for this separation. c. Determine the minimum reflux ratio. d. The applied reflux ratio is 3.76, and determine the boilup ratio (=V″/B) in the reboiler.

10

About 550 kmol/h of a binary mixture of water/acetic acid (70 mol% water) is separated by distillation in a bottom fraction with 98 mol% acetic acid and a

Exercises

55

distillate containing 5 mol% acetic acid. The relative volatility of water over acetic acid is 1.85. a. Calculate the amount of distillate and bottoms product this distillation yields? b. Determine graphically the minimum number of stages. The external reflux ratio R (=L′/D) is set at 3. The boilup ratio in the stripping section (=V″/B) is set to 10. c. Determine graphically the required number of equilibrium stages under these conditions. d. Determine graphically the optimal location for the feed stage. e. Determine graphically the slope of the feed line and from that the vapor fraction in the feed. 11

Isopropyl alcohol (IPA, boiling point 82.3 °C) needs to be separated from a 30 mol% IPA solution in water. The total IPA production in the plant amounts to 50 kmol/h and the maximum residual concentration in the water should be only 1 mol%. The feed consists of 90% of liquid and the efficiency of the trays is 90%. a. What is the maximum purity of the IPA that can be obtained by distillation under the current circumstances? b. Calculate the size of the bottoms and distillate stream to obtain an IPA purity of 70 mol%. c. Draw the yx-diagram including the pseudoequilibrium line. d. Determine the minimal reflux ratio. e. The real reflux ratio is 2.5, and determine the boilup ratio (=V″/B) in the reboiler. f. Determine graphically the number of real trays that is required for this distillation. VLE data (1 atm, mole fraction IPA) x 0.10 0.20 0.35 0.50 0.65 0.75 0.80 0.85 0.95 y 0.33 0.43 0.53 0.61 0.70 0.77 0.81 0.85 0.94

Chapter 3 Absorption and Stripping 3.1 Introduction In absorption (also called gas absorption, gas scrubbing and gas washing) a gas mixture is contacted with a liquid (the absorbent or solvent) to selectively dissolve one or more components by transfer from the gas to the liquid. The components transferred to the liquid are referred to as solutes or absorbates. The operation of absorption can be categorized on the basis of the nature of the interaction between absorbent and absorbate into the following three general types: 1. Physical solution. In this case, the component being absorbed is more soluble in the liquid absorbent than the other gases with which it is mixed but does not react chemically with the absorbent. As a result, the equilibrium concentration in the liquid phase is primarily a function of partial pressure in the gas phase and temperature. Examples are the drying of natural gas with diethylene glycol or the recovery of ethylene oxide and acrylonitrile with water from the reactor product stream. 2. Reversible reaction. This type of absorption is characterized by the occurrence of a chemical reaction between the gaseous component being absorbed and a component in the liquid phase to form a compound that exerts a significant vapor pressure of the absorbed component. The most important industrial example is the removal of acid gases (CO2 and H2 S) with mono- or diethanolamine solutions. 3. Irreversible reaction. In this case, a reaction occurs between the component being absorbed and a component in the liquid phase, which is essentially irreversible. Sulfuric acid and nitric acid production by SO3 and NO2 absorption in water is the most widely used example of this application. The use of physical absorption processes is usually preferred whenever feed gases are present in large amounts at high pressure and the amount of the component to be absorbed is relatively large. Chemical absorption usually has a much more favorable equilibrium relationship than physical absorption and is therefore often preferred when the components to be separated from feed gases are present in small concentrations and at low partial pressures. Gas absorption is usually carried out in vertical countercurrent columns as shown in Figure 3.1. The solvent is fed at the top whereas the gas mixture enters from the bottom. The absorbed substance is washed out by the lean solvent and leaves the absorber at the bottom as a liquid solution. Usually, the absorption column operates at a pressure higher than atmospheric pressure, taking advantage of the fact that gas solubility increases with pressure. The loaded solvent is often recovered in a subsequent stripping or desorption operation. After preheating, https://doi.org/10.1515/9783110654806-003

58

Chapter 3 Absorption and Stripping

Figure 3.1: Basic scheme of an absorption installation with stripping for regeneration.

the rich solvent is transported to the top of a desorption column that usually operates under lower pressure than in the absorption column. This second step is essentially the reverse of absorption in which the absorbate is removed from the solvent. Desorption can be achieved through a combination of methods: 1. flashing the solvent to lower the partial pressure of the dissolved components 2. reboiling the solvent to generate stripping vapor by evaporation of part of the solvent 3. stripping with an inert gas or steam Wide use is made of desorption by pressure reduction because the energy requirement are low. After depressurization, stripping with an inert gas is often more economic than thermal regeneration. However, because stripping is not perfect, the absorbent recycled to the absorber contains residual amounts of the absorbed solute. The desired purity of the absorbent determines the final costs of desorption. The necessary difference between the partial pressure of the absorbed key component over the regenerated solution and the purified gas serves as a criterion for determining the dimensions of the absorption and desorption equipment. The lean solvent, devoid of gas, flows through the heat exchanger, where part of the heat needed for heating the rich solvent is recovered, and then through a second heat exchanger, where it is cooled down to a desired temperature and flows into the absorption column. Usually a small amount of fresh solvent should be added to the column to replenish the solvent which was partly evaporated in the desorption column or underwent irreversible chemical reactions which take place in the whole system.

3.2 The aim of absorption

59

3.2 The aim of absorption Generally the commercial purpose of absorption processes can be divided into gas purification or product recovery, depending on whether the absorbed or the unabsorbed portion of the feed gas has the greater value. Typical gas purification applications are listed in Table 3.1. The removal of CO2 from hydrogen gas in ammonia production and the removal of acid gases (CO2 and H2 S) from natural gas are some of the most widespread applications that are being improved continuously by the development of new solvents, process configurations and design techniques. In both applications stripping is used for absorbent regeneration.

Table 3.1: Typical applications of absorption for gas purification. Impurity

Process

Absorbent

Ammonia

Indirect process (Coke oven gas) Ethanolamine

Water

Benfield

Potassium carbonate and activator in water Polyethylene glycol dimethyl ether Cuprous ammonium carbonate and formate in water Water Toluene Cyclohexane

Carbon dioxide and Hydrogen sulfide

Carbon monoxide

Selexol Copper ammonium salt

Hydrogen chloride Toluene Cyclohexane

Water wash Toluene scrubber Scrubber

Mono- or diethanolamine in water

Examples of absorption processes for product recovery are listed in Table 3.2. The absorption of SO3 and NOx in water to make concentrated sulfuric acid and nitric

Table 3.2: Typical applications of absorption for product recovery. Product

Process

Absorbent

Acetylene Acrylonitrile Maleic anhydride Melamine Nitric acid Sulfuric acid Urea

Steam cracking of hydrocarbons (Naphtha) Ammoxidation of propylene Butane oxidation Urea decomposition Ammonia oxidation (NOx absorption) Contact process (SO absorption) Synthesis (CO and NH absorption)

Dimethylformamide Water Water Water Water Water Ammonium carbamate solution

60

Chapter 3 Absorption and Stripping

acid, respectively, are probably the most widely used product recovery applications of absorption. Other frequently encountered examples are the recovery of various products from a gaseous product stream by inert absorbents such as water. In some cases the absorber is used as a reactor where the desired chemical compound is obtained by a liquid phase reaction of the absorbed gases. An illustration of such a process is the production of urea from CO2 and ammonia.

3.3 General design approach Both absorption and stripping can be operated as equilibrium stage operations with contact of liquid and vapor. The plate towers can be designed by following an adaptation of the McCabe–Thiele method. Packed towers can be designed by the use of HETP or preferably by mass transfer considerations. In both absorption and stripping, a separate phase is added as the separating agent. Thus the columns are simpler than those in distillation, in that reboilers and condensers are normally not used. Design or analysis of an absorber (or stripper) requires consideration of a number of factors, including the following: 1. Entering gas (liquid) flow rate, composition, temperature and pressure 2. Desired degree of recovery of one or more solutes 3. Choice of absorbent (stripping agent) 4. Operating pressure and temperature, and allowable gas pressure drop 5. Minimum absorbent (stripping agent) flow rate and actual absorbent (stripping agent) flow rate as a multiple of the minimum rate needed to make the separation 6. Number of equilibrium stages 7. Heat effects and need for cooling (heating) 8. Type of absorber (stripper) equipment 9. Height of absorber (stripper) 10. Diameter of absorber (stripper) The initial step in the design of the absorption system is selection of the absorbent and overall process to be employed. There is no simple analytical method for accomplishing this step. In most cases, more than one solvent can meet the process requirements, and the only satisfactory approach is an economic evaluation, which may involve the complete but preliminary design and cost estimate for more than one alternative. The ideal absorbent should: 1. have a high solubility for the solute(s) to minimize the need for absorbent 2. have a low volatility to reduce the loss of absorbent and facilitate separation of solute(s) 3. be stable to maximize absorbent life and reduce absorbent makeup requirement 4. be noncorrosive to permit use of common materials of construction

3.4 Absorption and stripping equilibria

61

5.

have a low viscosity to provide low pressure drop and high mass and heat transfer rates 6. be nonfoaming when contacted with the gas 7. be nontoxic and nonflammable to facilitate its safe use 8. be available, if possible within the process, or be inexpensive. The most widely used absorbents are water, hydrocarbon oils and aqueous solutions of acids and bases. For stripping the most common agents are water vapor, air, inert gases and hydrocarbon gases. Once an absorbent is selected, the design of the absorber requires the determination of basic physical property data such as density, viscosity, surface tension and heat capacity. The fundamental physical principles underlying the process of gas absorption are the solubility and heat of solution of the absorbed gas and the rate of mass transfer. Information on both must be available when sizing equipment for a given application. In addition to the fundamental design concepts based on solubility and mass transfer, many practical details have to be considered during actual plant design. The second step is the selection of the operating conditions and the type of contactor. In general, operating pressure should be high and temperature low for an absorber to minimize stage requirements and/or absorbent flow rate. Operating pressure should be low and temperature high for a stripper to minimize stage requirements or stripping agent flow rate. However, because maintenance of a vacuum is expensive, strippers are commonly operated at a pressure just above ambient. A high temperature can be used, but it should not be so high as to cause undesirable chemical reactions. Choice of the contactor may be done on the basis of system requirements and experience factors such as those discussed in Section 3.6. Following these decisions it is necessary to calculate material and heat balance calculations around the contactor, define the mass transfer requirements, determine the height of packing or number of trays and calculate contactor size to accommodate the liquid and gas flow rates with the selected column internals.

3.4 Absorption and stripping equilibria 3.4.1 Gaseous solute solubilities The most important physical property data required for the design of absorbers and strippers are gas–liquid equilibria. Since equilibrium represents the limiting condition for any gas–liquid contact, such data are needed to define the maximum gas purity and rich solution concentration attainable in absorbers, and the maximum lean solution purity attainable in strippers. Equilibrium data are also needed to establish the mass transfer driving force, which can be defined simply as the difference between the actual and equilibrium conditions at any point in a contactor. At

62

Chapter 3 Absorption and Stripping

equilibrium, a component of a gas in contact with a liquid has identical fugacities in both the gas and liquid phase. For ideal solutions Raoult’s law applies, see also eq. (2.6): yA =

PA0 xA Ptot

(3:1)

where yA is the mole fraction of A in the gas phase, Ptot is the total pressure, PA0 is the saturation pressure, the vapor pressure of pure A and xA is the mole fraction of A in the liquid. For nonideal mixtures Raoult’s law modifies into: yA =

0 γ∞ A PA xA Ptot

(3:2)

where γA∞ is the activity coefficient of solute A in the absorbent at infinite dilution. A more general way of expressing solubilities is through the dimensionless vapor–liquid distribution coefficient K, defined in the previous chapter: yA ≡ KA xA

(3:3)

Values of distribution coefficients are widely employed to represent hydrocarbon vapor–liquid equilibria in absorption and distillation calculations. Correlations and experimental data on the distribution coefficients of hydrocarbons are available from various sources. For moderately soluble gases with relatively little interaction between the gas and liquid molecules equilibrium data are usually represented by Henry’s law: pA = Ptot yA = HA xA

(3:4)

where pA is the partial pressure of A in the gas phase. HA is a Henry constant,1 which has the units of pressure. The Henry’s law constants depend upon temperature and usually follow an Arrhenius relationship:   − ΔHvap (3:5) HA = HA0 exp RT A plot of ln HA versus 1/T will then give a straight line. Usually a Henry constant increases with temperature, but is relatively independent of pressure at moderate levels. In general, for moderate temperatures, gas solubilities decrease with an increase in temperature. Henry’s constants for many gases and solvents are tabulated in various literature sources. Examples of Henry’s constants for a number of gases in pure water are given in Figure 3.2.

1 Note that we use HA = Henry coefficient of a component A, whereas ΔHvap = HV–HL refers to enthalpies.

63

3.4 Absorption and stripping equilibria

Figure 3.2: Solubilities of various gases in water expressed as the reciprocal of the Henry’s Law constant (adapted from [15]).

3.4.2 Minimum absorbent flow For each feed gas flow rate, absorbent composition, extent of solute absorption, operating pressure and operating temperature, a minimum absorbent flow rate exists that corresponds to an infinite number of countercurrent equilibrium contacts between the gas and liquid phases. As a result a tradeoff exists in every design problem between the number of equilibrium stages and the absorbent flow rates at rates greater than the minimum value. This minimum absorbent flow rate Lmin is obtained from a mass balance over the whole absorber, assuming equilibrium is obtained between incoming gas and outgoing absorbent liquid in the bottom of the column. An overall mass balance over the column illustrated by Figure 3.3 gives the following result: G yin + L xin = G yout + L xout

ð3:6Þ2

The lower the absorbent flow, the higher the xout. The theoretically highest possible concentration in the liquid xout determines the minimum absorbent flow, such as an infinitely long column where equilibrium can be assumed between incoming gas and outgoing liquid (xout = yin/K):

2 Note that in absorption the gas phase usually is denoted by G instead of V(apor) as in the previous chapter.

64

Chapter 3 Absorption and Stripping

Figure 3.3: Continuous, steady-state operation in a counter-current column.

Lmin = G ·

yin − yout yin − yout = G · yin xmax − xin K − xin

(3:7)

A similar derivation of the minimum stripping gas flow rate Gmin for a stripper results in an analogous expression: Gmin = L ·

xin − xout xin − xout =L · ymax − yin K xin − yin

(3:8)

3.5 Absorber and stripper design 3.5.1 Operating lines for absorption The McCabe–Thiele diagram is most useful when the operating line is straight. This requires that the energy balances are automatically satisfied and the ratio of liquid to vapor flow rate is constant. In order to have the energy balances automatically

3.5 Absorber and stripper design

65

satisfied, we must assume that the heat of absorption is negligible and the operation is isothermal. These two assumptions will guarantee satisfaction of the enthalpy balances. When the gas and liquid stream are both fairly dilute (say < 5%), the assumptions will probably be satisfied. We also desire a straight operating line at higher concentrations. This will automatically be true if we assume that the solvent is nonvolatile, the carrier gas is insoluble and define liquid and gas streams in terms of solute free solvent and carrier gas: L′ moles nonvolatile solvent=s = moles insoluble carrier gas=s G′

(3:9)

The results of these last two assumptions are that the mass balances for the solvent and carrier gas become ′ ′ ′ ′ ′ ′ LN = Ln = L0 = L′ = constant and GN + 1 = Gn = G1 = G′ = constant

(3:10)

Now overall flow rates of gas and liquid are not used because in more concentrated mixtures a significant amount of solute may be absorbed which would change gas and liquid flow rates. This would result in a curved operating line. Using L′ (=moles of nonvolatile solvent/s) and G′ (=moles insoluble carrier gas/s), we must define compositions in such a way that we can write a mass balance for solute B. The correct way to do this is to define the compositions as mole ratios: Y=

moles solute B in gas moles pure carrier gas

and X =

moles solute B in liquid moles pure solvent S

(3:11)

The mole ratios Y and X are related to the usual mole fractions by: Y=

y 1−y

and

X=

x 1−x

(3:12)

Substitution of X and Y into the ideal equilibrium expression, eq. (2.8), gives the equilibrium line expressed in mole ratios: Y =α·X

(3:13)

which represents a straight line through the origin. Note that both Y and X can be greater than unity. With mole ratio units we obtain for the gas and liquid stream leaving stage n: Yn G′ = and

moles B in gas stream moles carrier gas moles B in gas stream · = (3:14) moles carrier gas s s

66

Chapter 3 Absorption and Stripping

Xn L ′ =

moles B liquid stream moles solvent moles B liquid stream · = moles solvent s s

(3:15)

Thus we can easily write the steady-state mass balance, moles solute B in s−1 = moles solute B out s−1, in these units. The mass balance around the top of the column using the mass balance envelope shown in Figure 3.4 is Yn + 1 G′ + X0 L′ = Xn L′ + Y1 G′

(3:16)

Figure 3.4: Top section mass balance for an absorber.

Solving for Yn+1 we obtain: Yn + 1 =

L′ Xn + G′

  L′ X0 Y1 − G′

(3:17)

This is a straight line with slope L′/G′ and intercept Y1 – (L′/G′) · X0. It is the operating line for absorption. Thus if we plot ratios Y vs X we have a McCabe–Thiele type of graph as shown in Figure 3.5. The steps in the procedure are:

3.5 Absorber and stripper design

67

Figure 3.5: McCabe-Thiele diagram for absorption.

1. 2.

Plot Y vs X equilibrium data (convert from fractions to ratios) Values of X0, YN+1, Y1 and L′/G′ are known. Point (X0, Y1) is on the operating line since it represents passing streams; Xn follows from an overall balance. 3. Slope is L′/G′. Plot operating line. 4. Starting at stage 1, step off stages: equilibrium, operating line, equilibrium, etc.

Note that the operating line in absorption is above the equilibrium line, because solute is transferred from the gas to the liquid. In distillation we had material (the more volatile component) transferred from liquid to gas, and the operating line was below the equilibrium curve. The Y = X line has no significance in absorption. As usual the stages are counted at the equilibrium curve. The minimum L′/G′ ratio corresponds to a value of XN leaving the bottom of the tower in equilibrium with YN+1, the solute concentration in the feed gas. It takes an infinite number of stages for this equilibrium to be achieved. This minimal L′/G′ ratio can be derived from the McCabe–Thiele diagram as shown in Figure 3.6. The selection of the actual operating absorbent flow rate is based on some multiple of L′min, typically between 1.1 and 2. A value of 1.5 is usually close to the economically optimal conditions.

68

Chapter 3 Absorption and Stripping

Figure 3.6: Determination of the minimum (L/G) ratio for absorption from a McCabe–Thiele diagram.

3.5.2 Stripping analysis The number of equilibrium stages for the stripper is determined in a manner similar to that for absorption. Since stripping is very similar to absorption we expect a similar result. The mass balance for the column shown in Figure 3.7 is the same as for absorption and the operating line remains: Yn + 1

L′ Xn + = G′

  L′ X0 Y1 − G′

(3:18)

For stripping we know X0, XN, YN+1 and L′/G′, Y1 follows from an overall balance. Since (XN, YN+1) is a point on the operating line, we can plot the operating line and step off stages. This is illustrated in Figure 3.7. Note that the operating line is below the equilibrium curve because solute is transferred from liquid to gas. This is therefore similar to the stripping section of a distillation column. A maximum L′/G′ ratio can be defined. This corresponds to the minimum amount of stripping gas. Start from the known point (YN+1, XN) and draw a line to the intersection of X = X0 and the equilibrium curve. For a stripper, Y1 > YN+1 while the reverse is true in absorption. Thus the top of the column is on the right side in Figure 3.7 but on the left side in Figure 3.5.

3.5 Absorber and stripper design

69

Figure 3.7: McCabe–Thiele diagram for stripping.

3.5.3 Analytical Kremser solution McGabe–Thiele diagrams can be used to find graphically the number of equilibrium stages, no matter the operating and equilibrium expressions are linear or not. In case of linear and equilibrium lines, an analytical solution exists. When the solution is quite dilute (less than 1% in both gas and liquid), the total liquid and gas flow rates will not change significantly since only a small amount of solute is transferred. Now the entire analysis can be done with mole (or mass) fractions and constant total flow rates G and L. Alternatively, when the use of mole ratios and pure carrier gas and solvent flows result in linear operating and equilibrium lines, the following analysis applies equally well.3 In this case the column shown in Figure 3.8 will look like the one in Figure 3.4 with G′ = G and L′ = L. The mass balance around any stage n in terms of mole fractions and total flows: yn + 1 G + xn − 1 L = xn L + yn G

(3:19)

L yn + 1 − yn = G xn − xn − 1

(3:20)

which can be rewritten as

3 After conversion of mole fractions x, y into mole ratios X, Y and using L′, G′ instead of total flows L, G.

70

Chapter 3 Absorption and Stripping

x0 L

y1 G 1

n-1 xn-1

yn n

xn

yn+1 n+1

N yN+1 G

xN L

Figure 3.8: Coding of equilibrium stages, streams and fractions for the derivation of the Kremser equations.

It is essentially the same as eq. (3.16) except that the units are different. In case of the straight equilibrium line, that is, at low values of xn and yn, the equilibrium equation eq. (2.8) is reduced to yn = K xn

ð3:21Þ4

Equations (3.20) and (3.21) can be combined and, introducing the absorption factor A, rewritten as: A≡

L yn + 1 − yn yn + 1 − Kxn = = yn − Kxn − 1 K G Kðxn − xn − 1 Þ

(3:22)

Note that this absorption factor A is different from the separation factor as defined in eq. (1.1). For stage n–1 we find in an analogous way:

4 Here the equilibrium constant K replaces α = KA/KB in eq. (2.8).

3.5 Absorber and stripper design

yn − Kxn − 1 L = = A yn − 1 − Kxn − 2 KG

71

(3:23)

Elimination of xn-1 from eqs. (3.22) and (3.23) gives yn + 1 − Kxn = A2 yn − 1 − Kxn − 2

(3:24)

Continuing in a similar way we find for a cascade with N contacting equilibrium stages: yN + 1 − KxN yin − Kxout = = AN y1 − Kx0 yout − Kxin

(3:25)

Using the overall mass balance xout can be eliminated from eq. (3.25): xout = xin +

G G yin − yout L L

(3:26)

resulting in

 A

N

=

yin − Kxin yout − Kxin

   1 1 1− + A A

(3:27)

For specified values of yin, yout and xin eq. (3.27) is rewritten to calculate the number of equilibrium stages Nts:

   yin − Kxin 1 1 + ln 1− yout − Kxin A A (3:28) Nts = ln A This equation is known as the Kremser-equation for absorption. Equation (3.27) also allows calculating the fraction of the feed that is absorbed in a column with N equilibrium stages. Noting that yin–yout is the actual change in gas composition and yin–K·xin is the maximum possible change in composition, that is, if gas leaving is in equilibrium with entering liquid, then from eq. (3.27) the fraction absorbed fA can be derived: Fraction of a solute absorbed = fA =

yin − yout AN + 1 − A = N +1 yin − Kxin −1 A

(3:29)

Following an analogous derivation for stripping of a liquid by a gas, we are now interested in the change in liquid phase concentrations. Equation (3.25) is rewritten in terms of mole fractions solute in liquid: xin − yout =K 1 = SN = xout − yin =K AN

(3:30)

72

Chapter 3 Absorption and Stripping

where the solute stripping factor S is defined by S ≡ the overall mass balance provides yout = yin +

K·G L .

Elimination of yout with

L L xin − xout G G

(3:31)

resulting in

 Nts

S

=

xin − yin =K xout − yin =K



  1 1 1− + S S

(3:32)

For specified values of xin, xout and yin, the required number of equilibrium stages Nts is then calculated from the Kremser-equation for stripping: 

   xin − yin =K 1 1 1− + ln xout − yin =K S S (3:33) Nts = ln S Analogously to the fraction absorbed, the fraction of solute stripped, fS, is defined as the ration of the actual change in liquid composition and the maximum possible change. The degree of stripping obtained in a stripper with Nts equilibrium stages: Fraction of a solute stripped = fS =

xin − xout SNts + 1 − S = N +1 xin − yin =K S ts − 1

(3:34)

Values of L and G in moles per unit time may be taken as the entering values. Values of K depend mainly on temperature, pressure and liquid phase composition.

3.6 Industrial absorbers The main purpose of various industrial absorbers is to ensure large gas–liquid mass transfer area and to create such conditions that a high intensity of mass transfer is achieved. Although small-scale processes sometimes use batch-wise operation where the liquid is placed in the equipment and only gas is flowing, the continuous method of absorption is usually applied in large-scale industrial processes. There are various criteria for classifying absorbers. It seems that the best one is a widely used criterion that takes into account which of the phases (gas or liquid) is in a continuous or disperses form. Using this criterion, absorbers can be classified into the following groups: 1. Absorbers in which both phases are in a continuous form (packed columns) 2. Absorbers with a disperse gas phase and a continuous liquid phase (plate columns, bubble columns and packed bubbles columns) 3. Absorbers with a dispersed liquid phase and a continuous gas phase (spray columns)

3.6 Industrial absorbers

73

The industrially most frequently used absorbers are packed columns, plate towers, spray columns and bubble columns.

3.6.1 Packed columns Packed columns are the units most often used in absorption operations (Figure 3.9). Usually, they are cylindrical columns up to several meters in diameter and over 10 meters high. A large gas–liquid interface is achieved by introducing various packings into the column. The packing is placed on a support whose free cross section for gas flow should be at least equal to the packing porosity. Liquid is fed in at the top of the column and distributed over the packing through which it flows downwards. To guarantee a uniform liquid distribution over the cross section of the column, a liquid distributor is employed. Gas flows upward countercurrently to the falling liquid, which absorbs soluble species from the gas. The gas, which is not absorbed, flows away from the top of the column, usually through a mist eliminator. The mist eliminator separates liquid drops entrained by the gas from the packing. The separator may be a layer of the packing, mesh or it may be specially designed.

Figure 3.9: Schematic and operating principles of packed, tray and spray towers adapted from [18].

74

Chapter 3 Absorption and Stripping

Packings may be divided into two main groups: random and structured. The most popular random packing is rings. Raschig rings have a large specific packing surface and high porosity. They are hollow cylinders with an external diameter equal to the ring height that can be made of ceramic, metal, graphite or plastic. Recently, the application of structured packing in absorption has increased rapidly owing to the fact that it has a larger mass transfer area than random packing. Another advantage is that by using this packing it is possible, in contrast to random packing, to obtain the same values of mass transfer coefficient in the entire column.

3.6.2 Plate columns Another basic type of equipment widely applied in absorption processes is a plate column (Figure 3.9). The diameter and height of the column can reach 10 and 50 m, respectively, but usually they are much smaller. In plate columns various plate constructions guarantee good contact of the gas with the liquid. Taking into account the whole column, the flow of the gas with the liquid has a countercurrent character. Liquid is supplied to the highest plate, flows along it horizontally and, after reaching a weir, flows through a downcomer from plate to plate in cascade fashion. Gas is supplied below the lowest plate, and then it flows through perforations in the plate dispersers (e.g., holes in a sieve tray and slits in a bubble-cap tray) and bubbles through the flowing liquid at each plate. The application of such a flow pattern is aimed at ensuring the maximum mass transfer area and high turbulence of the gas and liquid phases, which results in obtaining high mass transfer coefficients in both phases. The distances between plates should be such that liquid droplets entrained by the gas are separated from the liquid and the gas is separated from the liquid in the downcomer. Usually, the plate-to-plate distance ranges from 0.2 to 0.6 m and depends mainly on the column diameter and the liquid load of the plate.

3.6.3 Spray and bubble columns In spray towers liquid is sprayed as fine droplets, which make contact with a cocurrently or countercurrently flowing gas (Figure 3.9). The gas and liquid flows are similar to plug flow. Bubble columns (Figure 3.10) are finding increasing application in processes when absorption is accompanied by a chemical reaction. They are also widely used as chemical reactors in processes where gas, liquid and solid phases are involved. In bubble columns the liquid is a continuous phase while the gas flows through it in the form of bubbles, the character of the gas flow is well represented by plug flow, while that of the liquid is between plug and ideally mixed flow. These absorbers can operate in cocurrent and countercurrent phase flow.

3.6 Industrial absorbers

Bubble column

75

Bubble column with Internals

Gas out

Gas out Liquid in

Liquid in

Gas in

Gas in

Liquid out

Liquid out

Figure 3.10: Schematic of a bubble column absorber, without and with internal packing.

3.6.4 Comparison of absorption columns In each apparatus, due to different hydrodynamic conditions, various values of the mass transfer coefficients occur in both phases. Therefore, when choosing a given type of absorber, the following criteria should be taken into account: 1. the required method of absorption (continuous or semi-continuous) 2. the flow rate of the gas and the liquid entering the absorber (e.g. high gas flow rate and low liquid flow rate need different types of equipment than in the case where the two flow rates commensurate) 3. the required liquid hold-up (large or small liquid hold-up is needed) 4. which phase controls mass transfer in the absorber (gas phase or liquid phase)? 5. is it necessary to remove heat from the absorber? 6. the corrosiveness of the absorption systems 7. the required size of the interface (large or small mass transfer area) 8. the physicochemical properties of the gas and the liquid (particularly viscosity and surface tension). Apart from these factors there are many others, which influence the selection of equipment. For instance, gas impurities or deposits formed during the absorption process require the application of a given absorber. Packed columns are preferable to tray columns for small installations, corrosive service, liquids with a tendency to foam, very high liquid-to-gas ratios and low pressure drop applications. In the handling of corrosive gases, packing but not plates,

76

Chapter 3 Absorption and Stripping

can be made from ceramic or plastic materials. Packed columns are also advantageous in vacuum applications because the pressure drop, especially for regularly structured packings, is usually less than in plate columns. In addition packed columns offer greater flexibility because the packing can be changed with relative ease to modify column-operating characteristics. Tray columns are particularly well suited for large installations and low-tomedium liquid flow rate applications. In general they offer a wider operating window for gas and liquid flow than a countercurrent packed column. That is, they can handle high gas flow rates and low liquid flow rates that would cause flooding in a packed column. Tray columns are also preferred in applications having large heat effects since cooling coils are more easily installed in plate towers and liquid can be withdrawn more easily from plates than from packings for external cooling. Furthermore they are advantageous for separations that require tall columns with a large number of transfer units because they are not subject to channeling of vapor and liquid streams which can cause problems in tall packed columns. The main disadvantages of tray columns are their high capital cost, especially when bubble-cap trays or special proprietary design are used, and their sensitivity to foaming. Spray contactors are used almost entirely for applications where pressure drop is critical, such as flue gas scrubbing. They are also useful for slurries that might plug packings or trays. Other important applications include particulate removal and hot gas quenching. When used for absorption, spray devices are not applicable to difficult separations because they are limited to only a few equilibrium stages even with countercurrent spray column designs. The low efficiency of spray columns is believed to be due to entrainment of droplets in the gas and back mixing of the gas induced by the sprays. The high energy consumed for atomizing liquid and liquid entrainment in the gas outlet stream are two additional important disadvantages. Bubble columns are particularly well suited for applications when significant liquid hold-up and long liquid residence time are required. An advantage of these columns is their relatively low investment cost, a large mass transfer area and high mass transfer coefficients in both phases. Disadvantages of bubble columns include a high-pressure drop of the gas and significant back mixing of the liquid phase. The latter disadvantage can be reduced by introducing an inert packing of high porosity to the bubble column. Such a packing eliminates to a large extent the effects of liquid phase mixing along the column height. In addition the packing may also cause an increase in the mass transfer surface area with relation to the bubble column at the same flows of both phases. The advantages of bubble columns appreciably exceed their disadvantages and therefore an increasing number of these columns are applied in industry.

Exercises

77

Nomenclature A f G H ΔHvap K KA L N p P Ptot R S T x, y X, Y α γ

L/K·G, absorption factor, eq. (.) Fraction absorbed or stripped (eqs. (.) and (.)) Gas flow Henry coefficient Molar heat of vaporization Equilibrium constant, eq. (.) Distribution coefficient, eq. (.) Liquid flow Number of (theoretical) stages Partial pressure Saturation pressure Total pressure Gas constant K·G/L, stripping factor, eq. (.) Temperature Mole fraction Mole ratio Selectivity, relative volatility and equilibrium constant Activity coefficient

– – mol s− – J mol− – – mol s− – N m− N m− N m− J mol− K− – K – – – –

Indices A n ts vap

Components Stage number Theoretical stages Evaporation

Exercises 1

A plate tower providing six equilibrium stages is employed for stripping ammonia from a waste water stream by means of countercurrent air at atmospheric pressure and 25 °C. Calculate the concentration of ammonia in the exit water if the inlet liquid concentration is 0.1 mole% ammonia in water, the inlet air is free of ammonia and 2,000 standard cubic meter (1 atm, 25 °C) of air are fed to the tower per m3 of waste water. The absorption equilibrium at 25 °C is given by the relation yNH3 = 1.414 xNH3. Mwater = 0.018 kg mol−1; ρwater = 1,000 kg m−3; R = 8.314 J mol−1 K−1; 1 atm = 1.01325 bar.

2

When molasses is fermented to produce a liquor containing ethanol, a CO2-rich vapor containing a small amount of ethanol is evolved. The alcohol can be recovered by absorption with water in a sieve tray tower. For the following conditions, determine the number of equilibrium stages required for countercurrent flow of

78

Chapter 3 Absorption and Stripping

liquid and gas if the liquid flow rate equals 1.5 times the minimum liquid flow rate, assuming isothermal, isobaric conditions in the tower and neglecting mass transfer of all components except ethanol. Entering gas:  kmol/h, % CO, % ethyl alcohol,  °C, . bar Entering absorbing liquid: % water,  °C, . bar Required ethanol recovery: % The vapor pressure of ethanol amounts 0.10 bar at 30 °C, and its liquid phase activity coefficient at infinite dilution in water can be taken as 7.5. 3

A gas stream consists of 90 mole% N2 and 10 mole% CO2. We wish to absorb the CO2 into water. The inlet water is pure and is at 5 °C. Because of cooling coils the operation can be assumed to be isothermal. Operation is at 10 bar. If the liquid flow rate is 1.5 times the minimum liquid flow rate, how many equilibrium stages are required to absorb 92% of the CO2? Choose a basis of 1 mole/ h of entering gas. The Henry coefficient of CO2 in water at 5 °C is 875 bar.

4

A vent gas stream in your chemical plant contains 15 wt% of a pollutant, the rest is air. The local authorities want to reduce the pollutant concentration to less than 1 wt%. You have decided to build an absorption tower using water as the absorbent. The inlet water is pure and at 30 °C. the operation is essentially isothermal. At 30 °C your laboratory has found that at low concentrations the equilibrium data can be approximated by y = 0.5·x (where y and x are weight fractions of the pollutant in vapor and liquid). Assume that air is not soluble in water and that water is nonvolatile. a. Find the minimum ratio of water to air (L′/G′)MIN on a solute-free basis b. With (L′/G′) = 1.22 (L′/G′)MIN find the total number of equilibrium stages and the outlet liquid concentration

5

A gas treatment plant often has both absorption and stripping columns as shown below. In this operation the solvent is continually recycled. The heat exchanger heats the saturated solvent, changing the equilibrium characteristics of the system so that the solvent can be stripped. A very common type of gas treatment plant is used for the drying of natural gas by physical absorption of water in a hygroscopic solvent such as diethylene glycol (DEG). In this case dry nitrogen is used as the stripping gas. a) At a temperature of 70 °C and a pressure of 40 bar the saturated vapor pressure of water is equal to 0.2 bar. It is known that water and DEG form a nearly ideal solution. Calculate the vapor–liquid equilibrium coefficient and draw the equilibrium line.

Exercises

79

b) Construct the operating line for xin = 0.02, yout = 0.0002 and L/G = 0.01. Determine the number of stages required to reduce the water molefraction from yin = 0.001 to yout = 0.0002. c) How many stages are required for L/G = 0.005. What happens for L/G = 0.004. Determine the minimal L/G ratio to obtain the desired separation. d) Desorption takes place at 120 °C and 1 bar. The saturated vapor pressure of water is equal to 2 bar. Construct the equilibrium and operating lines for desorption using yin = 0 and L/G = (L/G)max/1.5. xout and xin are to be taken from the absorber operating at L/G = 0.01. e) Calculate analytically the number of stages in both sections. f) Comment on the chosen value of the liquid mole fraction at the outlet of the absorber.

Schematic of natural gas absorptive drying operation

Chapter 4 General Design of Gas/Liquid Contactors 4.1 Introduction In the previous chapters, it was shown how to calculate the number of theoretical stages to separate a given feed with two components of different volatility into two fractions of predefined composition. Distillation is usually conducted in vertical cylindrical vessels that provide intimate contact between the rising vapor and the descending liquid. A distillation column normally contains internal devices for such an effective vapor–liquid contact. Their basic function is to provide efficient mass transfer between the two-phase vapor–liquid systems. However, the short contact times between the two phases in distillation does not allow complete establishment of thermodynamic equilibrium. The first aim of this chapter is to derive a tool to estimate the column height for a given separation. The number of real stages determines the height of the column. Therefore, we need to develop a model that translates the distance to the thermodynamic equilibrium to that number of real stages. The knowledge of the rate of interface mass transfer will appear to be very helpful in this matter. The second aim of this chapter is to estimate the minimum diameter of the column to be designed for a certain capacity. Both goals are achieved by modeling the vapor–liquid contact, starting from a kinetic analysis of mass transfer between the two phases involved. To understand what factors influence the rate of interface mass transfer, we first derive rate expressions for such mass transfer in idealized hydrodynamic environments. Although parts of the models derived below are applicable to liquid–liquid extraction, the examples will focus on two important devices for distillation, absorption and stripping: the plate column and the packed column.

4.2 Modeling mass transfer In gas–liquid contactors, two phases separated by an interface in between are present. Assume that a component A is diffusing from the gas phase to the liquid phase due to a difference in mole fraction as the driving force. Generally, diffusion as well as convection contributes to mass transport. Neglecting any other contribution, the rate of mass transfer of a component A at some distance z from the interface is given by ΦA = − DVAB Az ρV and https://doi.org/10.1515/9783110654806-004

d yA + yA ðΦA + ΦB Þ Az dz

(4:1a)

82

Chapter 4 General Design of Gas/Liquid Contactors

ΦA = − DLAB Az ρL

d xA + xA ðΦA + ΦB Þ Az dz

(4:1b)

where ϕA is the molar transfer rate of A (mol s–1); ρV, ρL are molar densities of vapor and liquid, respectively, (mol m–3); DVAB , DLAB are coefficients of molecular diffusion in vapor and liquid, respectively (m2 s–1); Az is the area at distance z from interface (m2); yA, xA is the mole fraction of A in vapor and liquid, respectively (–); z is the distance to interface (m). A mole fraction profile in case of spherical liquid or vapor droplets could be as shown in Figure 4.1a. In the stationary situation, the gradient shown is inversely proportional to the distance z squared.1 It is very reasonable to assume that there would be no resistance to transfer across the interface2; hence, yAi = K·xAi. Then, at low concentration (xA ≪ 1) or with equimolecular diffusion (ϕA = ϕB), the rate expressions given in eq. (4.1) reduce to d yA d xA L = − D A ρ (4:2) ΦA = − DVAB ALV ρV AB LV L d z z = 0 d z z = 0 where ALV is the interfacial area between liquid and vapor, comprising the total surface area of all droplets. This area depends on the droplet size, dp. For n vapor Mole fraction

Mole fraction a

yA

b

yA Convection + Diffusion + Eddies

yAi xAi

Convection + Diffusion + Eddies xA

Vapor

Liquid Distance z from interface

yAi Diffusion xAi xA Vapor

δV δL Distance z from interface

Liquid

Figure 4.1: Mole fraction profiles around spherical liquid or vapor droplets: (a) actual profiles and (b) profiles according to film theory.

1 See Exercise 4.1. 2 Except for very fast reactions at the interface, see R.B. Bird, W.E. Stewart & E.N. Lightfoot, Transport Phenomena, 2nd Edition, John Wiley & Sons, 2007.

4.2 Modeling mass transfer

83

droplets in a total volume V, the ratio of interfacial area ALV and a vapor volume3 εV·V reads (see also eq. (6.1)) n · πd2p ALV 6 = = εV · V n · π6 d3p dp

(4:3)

A useful simplification is achieved by adopting the film mass transfer theory to relate the driving force and the transfer rate. This theory is based on two assumptions (see Figure 4.1b): – a thin, laminar film exists at eitherside of the interface, and – in this film only molecular diffusion is taking place. Hence, according to this theory, no gradients exist in the bulk of the two phases. Resistance to mass transfer is concentrated in the boundary layers where only molecular diffusion occurs. The stationary state rate expressions according to the film theory are ΦA = kV ALV ρV ðyA − yAi Þ = kL ALV ρL ðxAi − xA Þ

(4:4)

where kV, kL is film or single-phase mass transfer coefficients (m s–1). When the boundary layers are very thin, the gradients in eq. (4.2) can be replaced by their difference quotients. Now, comparison with eq. (4.4) reveals that the mass transfer coefficient, k, equals the ratio of molecular diffusivity and film thickness, DAB/δ, where δ = Δz. Literature data of molecular diffusivities in the gas phase show values in the order of 0.1·10−4–10-4 m2 s–1, whereas mass transfer coefficients appear to be in the order of 0.01 m s–1. Thus, the order of magnitude of the film thickness in the gas phase, δV, is 10−3–10−2 m. Diffusivities in the liquid phase are much smaller in the order of 10−9 m2 s−1 and experimental values of liquid mass transfer coefficients are around 10−4 m s−1. Thus, the film thickness in the liquid phase, δL, is an order of magnitude smaller than that in the gas phase. In either case, the film thickness is sufficiently small to assume linear concentration gradients. The simplified picture according to the film theory is shown in Figure 4.1b. This gives only a rough, inaccurate model of transport through liquid vapor interfaces. Nevertheless, the film theory appears to be an adequate tool for many engineering applications, especially when it comes to understanding principles of mass transfer in absorption or distillation towers. The interface compositions xAi and yAi, shown in Figure 4.1, are connected through eq. (4.4), which can be rearranged to show as a linear operating line in a y–x diagram:

3 εV = vapor volume fraction.

84

Chapter 4 General Design of Gas/Liquid Contactors

yA − yAi = −

kL ρL ðxA − xAi Þ kV ρV

(4:5)

The slope of this straight line depends on the ratio of the two mass transfer coefficients involved. The absence of mass transport resistance in the interface suggests that both interface compositions are in equilibrium. This assumption gives the second equation required to calculate the interface compositions. A graphical method to do so is presented in Figure 4.2. y yA

a

y

kL ρL Slope – — — kV ρV

b

y

c kV > kL yA

yA = yAi

yAi

yAi Equilibrium line

xA

xAi

Equilibrium line

Equilibrium line

x

xA

xAi

x

xA = xAi

x

Figure 4.2: Driving force and interface compositions. (a) Transport in both liquid and vapor determines the overall transport rate; (b) transport in vapor is very fast; and (c) transport in liquid is very fast.

Equation (4.5) and the equilibrium curve are plotted in a y–x graph. The intersection of the straight line with the equilibrium curve gives the values of xAi and yAi. Two limiting situations can be considered. When the transport in the gas phase is very fast compared to that in the liquid phase, the gradient across the gas film can be neglected to that across the liquid film. All resistance to mass transport is said to be in the liquid phase; this situation is depicted in Figure 4.2b. Figure 4.2c shows the opposite situation where the resistance is entirely in the gas phase. The application of the above method to model transfer rates in G/L contactors would be much easier to handle when the interface compositions could be eliminated from the driving force factor. The following mathematical treatment will do just that.4 For the sake of simplicity, we assume a linear equilibrium expression: y*A = K · xA

(4:6)

where y*A is a hypothetical gas phase composition in equilibrium with the actual mole fraction xA in the liquid phase (see diagrams page 85). Combination with eq. (4.4) gives

4 See also Exercise 4.2.

4.2 Modeling mass transfer

ΦA K · ΦA and K · xAi − K · xA = yAi − y*A = kV ALV ρV kL ALV ρL

yA − yAi =

85

(4:7)

Summation of both expressions eliminates yAi giving (see eq. 4.8) ΦA = kOV ALV ρV ðyA − y*A Þ

(4:8)

Mole fraction xi x y y – y*

yi =

Kxi y*= Kx

Vapor

Liquid

(see Eq. 4.8)

with kOV =

kV 1+

kV kL

·

K ρV ρL

=

kV 1+

mkV kL

  kOV = overall mass transfer coefficient, based on the gas phase ms − 1 m=

K ρV ρL

, volumetric distribution coefficient (see eq. 4.9)

x* = y/K

Mole fraction

x* – x

xi = yi/K

x

y yi

Vapor

Liquid

(see Eq. 4.9)

Similarly, elimination of xAi from eq. (4.4) gives a rate expression with an overall mass transfer coefficient kOL, applicable when the main resistance against mass transfer is in the liquid phase: ΦA = kOL ALV ρL ðx*A − xA Þ

(4:9)

86

Chapter 4 General Design of Gas/Liquid Contactors

with summarizing, the rate of mass transfer across a phase boundary depends on – the driving force, yA − y*A or xA* − xA , the asterisk * referring to a hypothetical equilibrium; – the contact area ALV between the two phases; and – an overall mass transfer coefficient, kOV or kOL. The maximum mass transfer rate is obtained when these three factors on the righthand side of eqs. (4.8) or (4.9) are as large as possible: – The largest driving force is achieved when the overall flow pattern allows countercurrent contact between equal streams of gas and liquid without significant remixing. An important condition is that both phases are distributed uniformly over the entire flow area. – A large contact area ALV is desirable for mass transfer and mainly determined by the used column internals. Depending on the type of internal devices used, the contacting may occur in discrete steps, called plates or trays, or in a continuous differential manner on the surface of a packing material. Tray columns and packed columns are most often used for distillation since they guarantee excellent countercurrent flow and permit a large overall height. The internals provide a large mass transfer area. In tray columns, mass transfer areas range from 30 to 100 m2 per unit tray area,5 and in packed columns from 200 to 500 m2 per unit column volume. – The mass transfer coefficient increases proportionally with the relative velocities between the liquid and vapor phases and is also improved by constant regeneration of the contact area between the phases. It is difficult to measure the overall mass transfer coefficients as such. The usual experimental setups designed to determine mass transport under well-defined conditions (temperature, pressure, concentration, hydrodynamic flow pattern) produce data to calculate the product of mass transfer coefficient and contact area. In the following sections, L/G mass transfer will be modeled in tray – or plate – columns as well as in packed columns.

4.3 Plate columns Figure 4.3 shows the most important features of a tray column. The gas flows upward within the column through perforations in horizontal trays, and the condensed liquid

5 Specific interfacial area ALV in gas–liquid systems may be estimated from gas volume fraction ε and bubble diameter d as ALV = 6ε/d in m2 m–3 (see eq. (4.3), also eq. (6.2) for interfacial area in cylinders).

4.3 Plate columns

87

Vapor to condenser

Reflux from condenser

d

a

c d

b

e

Liquid feed

f

c g h Vapor from reboiler

Liquid to reboiler

Figure 4.3: Cutaway section of a plate column: a, downcomer; b, tray support; c, sieve trays; d, man way; e, outlet weir; f, inlet weir; g, side wall of downcomer; and h, liquid seal. Reproduced with permission from [26].

flows countercurrently downward. However, as indicated in Figure 4.4, the two phases exhibit cross-flow to each other on the individual trays. The liquid enters the cross-flow plate from the bottom of the downcomer belonging to the plate above and flows across the perforated active or bubbling area. The ascending gas from the plate below passes through the perforations and aerates the liquid to form a large interfacial area between the two phases. It is in this zone where the main vapor–liquid mass transfer occurs. The vapor subsequently disengages from the aerated mass on the plate and rises to the tray above. The aerated liquid flows over the exit weir into a downcomer, where most of the trapped vapor escapes from the liquid and flows back to the interplate vapor space. Some of the liquid accumulates in each downcomer to compensate for the pressure drop caused by the gas as it passes through the tray. The liquid then leaves the plate by flowing through the downcomer outlet onto the tray below. In large diameter cross-flow plates, multiple liquid flow path

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Chapter 4 General Design of Gas/Liquid Contactors

Liquid deentraining zone

Vapor disengaging from liquid

Downcomer Exit weir

Downcomer clearance

Degassed liquid

Bubbling area

Figure 4.4: Schematic of flow pattern in a cross-flow plate distillation column.

plates with multiple downcomers are used to prevent the hydraulic gradient of the liquid flowing across the plate becomes excessive. Three principal vapor–liquid contacting devices are used for cross-flow plate design: the sieve plate, the valve plate and the bubble cap plate (see Figure 4.5): – Bubble cap plates have been used almost exclusively in the chemical industry until the early 1950s. As shown in Figure 4.5, their design prevents liquid from leaking downward through the tray. The vapor flows through a hole in the plate floor, through the riser, reverses direction in the dome of the cap, flows downward and exits through the slots in the cap. However, the complex bubble caps are relatively expensive and have a higher pressure drop than the former two designs. This limits their usage in newer installations to low liquid flow rate applications or to those cases where the widest possible operating range is desired. – Sieve plates have become very important because they are simple, inexpensive, have high separation efficiency and produce a low pressure drop across the tray. Conventional sieve plates contain typically 1–12 mm holes and exhibit ratios of open area to active area ranging from 1:20 to 1:7. If the open area is too small, the pressure drop across the plate is excessive, if the open area is too large the liquid weeps or dumps through the holes. – Valve plates are a relatively new development that represents a variation of the sieve plate with liftable valve units such as those shown in Figure 4.6 fitted in the holes. The liftable valves prevent the liquid from leaking at low gas loads

4.3 Plate columns

Liquid

89

Bubble cap tray

Vapor

Liquid

Sieve tray

Liquid

Vapor

Valve tray

Vapor

Figure 4.5: Schematic of a bubble cap, sieve and valve tray.

Koch floating valve

Nutter floating valve

Sulzer fixed valve

Figure 4.6: Examples of valves used in valve plates.

and to avoid excessive pressure increase at high gas loads. The main advantage is the ability to maintain efficient operation while being able to vary the gas load up to a factor of 4–5. This capability gives valve trays a much larger operational flexibility than any other tray design.

4.3.1 Dimensioning a tray column Product specification of a distillation, absorption or stripping process includes at least product purity and capacity. In the previous chapters, methods are discussed to determine the number of theoretical stages required to achieve the desired

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Chapter 4 General Design of Gas/Liquid Contactors

product purity. This relates to the height of the column, the more trays are required, the higher the column. The capacity of the process determines the diameter of the column. This section focuses on methods to estimate the minimum dimensions of a tray column: the minimum column height necessary to meet the desired product purity and the minimum diameter necessary to accommodate the required capacity (at constant pressure, see Figure 2.3).

4.3.2 Height of a tray column Although the column requirements are calculated in terms of theoretical or equilibrium number of stages, Nts, the design must specify the actual number of stages, Ns. The minimum height of a column is then calculated from the minimum distance Hspacing between adjacent trays and the number of actual stages: Hcolumn = Hspacing · Ns

(4:10)

Thus, an estimate of column height involves determination of the number of real stages, Ns, and the tray spacing, Hspacing. 4.3.2.1 Tray spacing The distance between two trays should be as small as possible but sufficiently large to prevent liquid droplets to reach the tray above. In industrial tray columns, spacing from 0.15 to 1.0 m is used. This value depends on vapor load and tray diameter. In columns with a diameter of 1 m or larger, a typical tray spacing of 0.3–0.6 m is found. A value of 0.5 m appears to be a reasonable initial estimate. In a detailed tray design, this value is subject to revision; however, that part of the design is outside the scope of this textbook. 4.3.2.2 Tray efficiency Due to relatively high flow rates in gas–liquid towers, contact times on trays are not sufficient to establish thermodynamic equilibrium. A measure of the performance of a real tray in approaching equilibrium is the efficiency of a real tray. In the previous chapters, methods were discussed to determine the total number of theoretical stages for a given separation such as distillation, absorption or stripping. The overall efficiency Eo of a column is defined by the number of theoretical trays, Nts, divided by the actual number of trays required to obtain the desired product purity.6

6 In distillation where the reboiler counts as one theoretical stage, this definition should be adjusted accordingly.

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4.3 Plate columns

Eo ≡

Nts Ns

(4:11)

Most hydrocarbon distillation systems in commercial columns achieve overall tray efficiencies of 60–80%7; in absorption processes the range is 10–50%. These figures are just guidelines. They have no theoretical basis and are not suitable for design purposes. A more fundamental approach would be to derive an expression for the efficiency of a single plate in terms of local driving force for interphase mass transport and hydrodynamic conditions. First, we need a hydrodynamic model of a tray. A reasonable model would be to assume that the gas phase, passing at high velocity through the liquid layer on a tray, behaves as a plug flow. Further that its concentration changes only in the axial direction, and radial concentration gradients can be neglected. The gas flow causes extensive mixing in the liquid layer, so the liquid is assumed being ideally mixed. The composition yn in gas phase leaving the nth tray is independent of the radial position. Now the tray, plate or Murphree efficiency EMV compares the actual difference yn + 1 − yn , leaving two consecutive trays to the maximum achievable difference yn + 1 − y*n : EMV =

yn + 1 − yn yn + 1 − y*n

(4:12)

Figure 4.7 gives an amplification of the tray efficiency in an absorber. Here, the lighter component in the gas phase from tray n + 1 is absorbed by the liquid, which has the same composition xn everywhere on the tray. At increasing height in the V.y

a

b

ALV

y yn+1

L . x n–1 xn

y y +Δy

Equilibrium line

yn

A +ΔA

yn*

A

V . ( y+ Δy)

yn* = K x n koy.ρy(y–yn*).ΔA

V.y Xn

xn

yn* = K x n

0

L.xn

x V.yn+1 Figure 4.7: Mass transport on a model nonequilibrium tray in absorber: (a) distance to equilibrium and (b) elements in mass balance for interphase mass transport.

7 Eo sometimes is expressed as a percentage by multiplication with 100.

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Chapter 4 General Design of Gas/Liquid Contactors

liquid layer, the mole fraction y decreases. On an equilibrium tray the composition of the gas phase leaving tray n would equal y*n , whereas on a real tray the exit value amounts to yn. In the following paragraph, the concept of transfer units is used to estimate the number of real trays, Ns. 4.3.2.3 Transfer units The next step is to relate the mole fractions in eq. (4.12) to kinetic parameters and flow rates. The hydrodynamic simplification for a plate defined above (liquid ideally mixed, plug flow in gas phase) allows setting up a mass balance in a thin liquid layer at a certain position h above the tray. Instead of height h, it is convenient to take the interphase ALV present on the tray as independent parameter. A thin liquid layer with thickness Δh comprises ΔA unit surface area. The driving force for interphase transport at this position equals y − y*n . Note that y*n , the virtual gas phase composition in equilibrium with the liquid of composition xn, is constant. In the stationary situation, the difference in molar flow of the absorbable component in the gas phase is balanced by interphase transport (see Figure 4.7). Applying the rate expression given in eq. (4.8), the mass balance becomes V · y = V · ðy + ΔyÞ + kOV · ρV · ðy − y*n Þ · ΔA

(4:13)

For the sake of clarity, we restrict ourselves to the case that heat effects can be neglected and that the molar vapor flow rate, V, is constant, for example, equal molal overflow and/or the concentration is low (